Dirichlet series
| L(s) = 1 | + 30·4-s + 30·9-s + 64·11-s + 403·16-s + 80·29-s + 900·36-s + 1.92e3·44-s − 7·49-s + 3.07e3·64-s + 376·71-s − 352·79-s + 495·81-s + 1.92e3·99-s − 636·109-s + 2.40e3·116-s + 968·121-s + 127-s + 131-s + 137-s + 139-s + 1.20e4·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 842·169-s + ⋯ |
| L(s) = 1 | + 15/2·4-s + 10/3·9-s + 5.81·11-s + 25.1·16-s + 2.75·29-s + 25·36-s + 43.6·44-s − 1/7·49-s + 48.0·64-s + 5.29·71-s − 4.45·79-s + 55/9·81-s + 19.3·99-s − 5.83·109-s + 20.6·116-s + 8·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 83.9·144-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 4.98·169-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 5^{40} \cdot 7^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 5^{40} \cdot 7^{20}\right)^{s/2} \, \Gamma_{\C}(s+1)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(3^{20} \cdot 5^{40} \cdot 7^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.28791\times 10^{23}\) |
| Root analytic conductor: | \(3.78222\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 3^{20} \cdot 5^{40} \cdot 7^{20} ,\ ( \ : [1]^{20} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(4114.912718\) |
| \(L(\frac12)\) | \(\approx\) | \(4114.912718\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - p T^{2} )^{10} \) |
| 5 | \( 1 \) | |
| 7 | \( 1 + p T^{2} + 6429 T^{4} + 21732 T^{6} + 505314 p^{2} T^{8} + 29418 p^{4} T^{10} + 505314 p^{6} T^{12} + 21732 p^{8} T^{14} + 6429 p^{12} T^{16} + p^{17} T^{18} + p^{20} T^{20} \) | |
| good | 2 | \( ( 1 - 15 T^{2} + 17 p^{3} T^{4} - 907 T^{6} + 4789 T^{8} - 5257 p^{2} T^{10} + 4789 p^{4} T^{12} - 907 p^{8} T^{14} + 17 p^{15} T^{16} - 15 p^{16} T^{18} + p^{20} T^{20} )^{2} \) |
| 11 | \( ( 1 - 16 T + 398 T^{2} - 3518 T^{3} + 57509 T^{4} - 374444 T^{5} + 57509 p^{2} T^{6} - 3518 p^{4} T^{7} + 398 p^{6} T^{8} - 16 p^{8} T^{9} + p^{10} T^{10} )^{4} \) | |
| 13 | \( ( 1 + 421 T^{2} + 79836 T^{4} + 981039 p T^{6} + 2035880823 T^{8} + 307536538872 T^{10} + 2035880823 p^{4} T^{12} + 981039 p^{9} T^{14} + 79836 p^{12} T^{16} + 421 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 17 | \( ( 1 + 592 T^{2} + 146646 T^{4} + 22775706 T^{6} + 11523048585 T^{8} + 4833069478380 T^{10} + 11523048585 p^{4} T^{12} + 22775706 p^{8} T^{14} + 146646 p^{12} T^{16} + 592 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 19 | \( ( 1 - 1447 T^{2} + 1050165 T^{4} - 554582916 T^{6} + 245616636570 T^{8} - 94582670333418 T^{10} + 245616636570 p^{4} T^{12} - 554582916 p^{8} T^{14} + 1050165 p^{12} T^{16} - 1447 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 23 | \( ( 1 - 1950 T^{2} + 2083027 T^{4} - 1642650556 T^{6} + 1130054729701 T^{8} - 656990798309146 T^{10} + 1130054729701 p^{4} T^{12} - 1642650556 p^{8} T^{14} + 2083027 p^{12} T^{16} - 1950 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 29 | \( ( 1 - 20 T + 2507 T^{2} - 55402 T^{3} + 3313889 T^{4} - 67970602 T^{5} + 3313889 p^{2} T^{6} - 55402 p^{4} T^{7} + 2507 p^{6} T^{8} - 20 p^{8} T^{9} + p^{10} T^{10} )^{4} \) | |
| 31 | \( ( 1 - 6277 T^{2} + 19393164 T^{4} - 39088975347 T^{6} + 56994871213239 T^{8} - 62699896233614232 T^{10} + 56994871213239 p^{4} T^{12} - 39088975347 p^{8} T^{14} + 19393164 p^{12} T^{16} - 6277 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 37 | \( ( 1 - 9548 T^{2} + 44216694 T^{4} - 131713023018 T^{6} + 7556518324197 p T^{8} - 441256561463128116 T^{10} + 7556518324197 p^{5} T^{12} - 131713023018 p^{8} T^{14} + 44216694 p^{12} T^{16} - 9548 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 41 | \( ( 1 - 10156 T^{2} + 51214350 T^{4} - 169530757398 T^{6} + 412496698272513 T^{8} - 779908516768124700 T^{10} + 412496698272513 p^{4} T^{12} - 169530757398 p^{8} T^{14} + 51214350 p^{12} T^{16} - 10156 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 43 | \( ( 1 - 8861 T^{2} + 38625087 T^{4} - 108514994670 T^{6} + 231815383369293 T^{8} - 437532221283330075 T^{10} + 231815383369293 p^{4} T^{12} - 108514994670 p^{8} T^{14} + 38625087 p^{12} T^{16} - 8861 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 47 | \( ( 1 + 9766 T^{2} + 53716221 T^{4} + 206379165672 T^{6} + 620986862825538 T^{8} + 1510406891680637412 T^{10} + 620986862825538 p^{4} T^{12} + 206379165672 p^{8} T^{14} + 53716221 p^{12} T^{16} + 9766 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 53 | \( ( 1 - 16908 T^{2} + 144304774 T^{4} - 820703678302 T^{6} + 3432867026880313 T^{8} - 10975084604115414124 T^{10} + 3432867026880313 p^{4} T^{12} - 820703678302 p^{8} T^{14} + 144304774 p^{12} T^{16} - 16908 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 59 | \( ( 1 - 10072 T^{2} + 63609438 T^{4} - 320162260506 T^{6} + 1328454263582721 T^{8} - 4786725393186808092 T^{10} + 1328454263582721 p^{4} T^{12} - 320162260506 p^{8} T^{14} + 63609438 p^{12} T^{16} - 10072 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 61 | \( ( 1 - 20197 T^{2} + 183312108 T^{4} - 1047496391811 T^{6} + 4605604707575319 T^{8} - 17839432563323310360 T^{10} + 4605604707575319 p^{4} T^{12} - 1047496391811 p^{8} T^{14} + 183312108 p^{12} T^{16} - 20197 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 67 | \( ( 1 - 21767 T^{2} + 272009760 T^{4} - 2357870661189 T^{6} + 15344973213250563 T^{8} - 77826309200282160576 T^{10} + 15344973213250563 p^{4} T^{12} - 2357870661189 p^{8} T^{14} + 272009760 p^{12} T^{16} - 21767 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 71 | \( ( 1 - 94 T + 17426 T^{2} - 895868 T^{3} + 116598785 T^{4} - 4514845052 T^{5} + 116598785 p^{2} T^{6} - 895868 p^{4} T^{7} + 17426 p^{6} T^{8} - 94 p^{8} T^{9} + p^{10} T^{10} )^{4} \) | |
| 73 | \( ( 1 + 19282 T^{2} + 253351965 T^{4} + 2388835550904 T^{6} + 17610142623434370 T^{8} + \)\(10\!\cdots\!44\)\( T^{10} + 17610142623434370 p^{4} T^{12} + 2388835550904 p^{8} T^{14} + 253351965 p^{12} T^{16} + 19282 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 79 | \( ( 1 + 88 T + 19074 T^{2} + 716634 T^{3} + 120361089 T^{4} + 1817536980 T^{5} + 120361089 p^{2} T^{6} + 716634 p^{4} T^{7} + 19074 p^{6} T^{8} + 88 p^{8} T^{9} + p^{10} T^{10} )^{4} \) | |
| 83 | \( ( 1 + 42952 T^{2} + 805000830 T^{4} + 8743622211498 T^{6} + 65519412158264673 T^{8} + \)\(43\!\cdots\!84\)\( T^{10} + 65519412158264673 p^{4} T^{12} + 8743622211498 p^{8} T^{14} + 805000830 p^{12} T^{16} + 42952 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 89 | \( ( 1 + 1622 T^{2} + 175206285 T^{4} + 194652778248 T^{6} + 15357008279584338 T^{8} + 9462869862475380804 T^{10} + 15357008279584338 p^{4} T^{12} + 194652778248 p^{8} T^{14} + 175206285 p^{12} T^{16} + 1622 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 97 | \( ( 1 + 45319 T^{2} + 930044589 T^{4} + 11414926232580 T^{6} + 98402520040038354 T^{8} + \)\(81\!\cdots\!66\)\( T^{10} + 98402520040038354 p^{4} T^{12} + 11414926232580 p^{8} T^{14} + 930044589 p^{12} T^{16} + 45319 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.31328356803508094748923820853, −2.27963720326015011022594783282, −2.23809781966233004213505146416, −1.99712380764031405441528305480, −1.92650251913919211236187759466, −1.85073191811617811810006395236, −1.72517493943748805474436117668, −1.61522142757034690985008456107, −1.55077970341676650661681366291, −1.51414655187507302736201490513, −1.43156272409451355703989309785, −1.42604143231268070775459101771, −1.40982899061680582381040450877, −1.34172831851999612827413713633, −1.33298998664118673314226897274, −1.26968201909701496660472928371, −1.18884984689322957650878174351, −1.14879185296354926653358351963, −0.925782528735761886115064727806, −0.76895395686370796321759707442, −0.70725419671322374761039586523, −0.37888260443250834916880923764, −0.36424861537198370538785266491, −0.31528373393825391011737454067, −0.30698043506121295815021812638, 0.30698043506121295815021812638, 0.31528373393825391011737454067, 0.36424861537198370538785266491, 0.37888260443250834916880923764, 0.70725419671322374761039586523, 0.76895395686370796321759707442, 0.925782528735761886115064727806, 1.14879185296354926653358351963, 1.18884984689322957650878174351, 1.26968201909701496660472928371, 1.33298998664118673314226897274, 1.34172831851999612827413713633, 1.40982899061680582381040450877, 1.42604143231268070775459101771, 1.43156272409451355703989309785, 1.51414655187507302736201490513, 1.55077970341676650661681366291, 1.61522142757034690985008456107, 1.72517493943748805474436117668, 1.85073191811617811810006395236, 1.92650251913919211236187759466, 1.99712380764031405441528305480, 2.23809781966233004213505146416, 2.27963720326015011022594783282, 2.31328356803508094748923820853
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.