Properties

Label 16-384e8-1.1-c1e8-0-1
Degree $16$
Conductor $4.728\times 10^{20}$
Sign $1$
Analytic cond. $7813.87$
Root an. cond. $1.75107$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·11-s + 8·19-s + 16·29-s + 24·31-s + 16·37-s + 8·43-s + 24·49-s − 16·53-s − 32·59-s − 16·61-s + 16·67-s − 24·79-s − 2·81-s + 40·83-s + 32·107-s + 16·113-s + 32·121-s − 16·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 2.41·11-s + 1.83·19-s + 2.97·29-s + 4.31·31-s + 2.63·37-s + 1.21·43-s + 24/7·49-s − 2.19·53-s − 4.16·59-s − 2.04·61-s + 1.95·67-s − 2.70·79-s − 2/9·81-s + 4.39·83-s + 3.09·107-s + 1.50·113-s + 2.90·121-s − 1.43·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{56} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(7813.87\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{56} \cdot 3^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(7.960904475\)
\(L(\frac12)\) \(\approx\) \(7.960904475\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( ( 1 + T^{4} )^{2} \)
good5 \( 1 + 16 T^{3} - 12 T^{4} - 48 T^{5} + 128 T^{6} + 32 T^{7} - 506 T^{8} + 32 p T^{9} + 128 p^{2} T^{10} - 48 p^{3} T^{11} - 12 p^{4} T^{12} + 16 p^{5} T^{13} + p^{8} T^{16} \)
7 \( 1 - 24 T^{2} + 292 T^{4} - 2440 T^{6} + 17222 T^{8} - 2440 p^{2} T^{10} + 292 p^{4} T^{12} - 24 p^{6} T^{14} + p^{8} T^{16} \)
11 \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + 12 p T^{4} - 344 T^{5} + 2400 T^{6} - 13000 T^{7} + 54374 T^{8} - 13000 p T^{9} + 2400 p^{2} T^{10} - 344 p^{3} T^{11} + 12 p^{5} T^{12} - 8 p^{6} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \)
13 \( 1 + 64 T^{3} - 4 T^{4} - 704 T^{5} + 2048 T^{6} + 1408 T^{7} - 53466 T^{8} + 1408 p T^{9} + 2048 p^{2} T^{10} - 704 p^{3} T^{11} - 4 p^{4} T^{12} + 64 p^{5} T^{13} + p^{8} T^{16} \)
17 \( ( 1 + 36 T^{2} + 64 T^{3} + 662 T^{4} + 64 p T^{5} + 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
19 \( 1 - 8 T + 32 T^{2} - 120 T^{3} + 452 T^{4} - 2168 T^{5} + 10080 T^{6} - 37832 T^{7} + 138918 T^{8} - 37832 p T^{9} + 10080 p^{2} T^{10} - 2168 p^{3} T^{11} + 452 p^{4} T^{12} - 120 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \)
23 \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{4} \)
29 \( 1 - 16 T + 128 T^{2} - 32 p T^{3} + 6580 T^{4} - 38208 T^{5} + 199680 T^{6} - 1073680 T^{7} + 5802054 T^{8} - 1073680 p T^{9} + 199680 p^{2} T^{10} - 38208 p^{3} T^{11} + 6580 p^{4} T^{12} - 32 p^{6} T^{13} + 128 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \)
31 \( ( 1 - 12 T + 164 T^{2} - 1140 T^{3} + 8218 T^{4} - 1140 p T^{5} + 164 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( 1 - 16 T + 128 T^{2} - 1008 T^{3} + 5948 T^{4} - 15248 T^{5} - 9344 T^{6} + 717840 T^{7} - 7530650 T^{8} + 717840 p T^{9} - 9344 p^{2} T^{10} - 15248 p^{3} T^{11} + 5948 p^{4} T^{12} - 1008 p^{5} T^{13} + 128 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \)
41 \( 1 - 200 T^{2} + 19452 T^{4} - 1244536 T^{6} + 58583750 T^{8} - 1244536 p^{2} T^{10} + 19452 p^{4} T^{12} - 200 p^{6} T^{14} + p^{8} T^{16} \)
43 \( 1 - 8 T + 32 T^{2} - 56 T^{3} + 260 T^{4} - 504 T^{5} - 2720 T^{6} + 625528 T^{7} - 7635866 T^{8} + 625528 p T^{9} - 2720 p^{2} T^{10} - 504 p^{3} T^{11} + 260 p^{4} T^{12} - 56 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \)
47 \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{4} \)
53 \( 1 + 16 T + 128 T^{2} + 928 T^{3} + 8564 T^{4} + 82496 T^{5} + 654336 T^{6} + 5021328 T^{7} + 38116486 T^{8} + 5021328 p T^{9} + 654336 p^{2} T^{10} + 82496 p^{3} T^{11} + 8564 p^{4} T^{12} + 928 p^{5} T^{13} + 128 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \)
59 \( ( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \)
61 \( 1 + 16 T + 128 T^{2} + 1392 T^{3} + 14204 T^{4} + 79760 T^{5} + 426880 T^{6} + 2945904 T^{7} + 19569574 T^{8} + 2945904 p T^{9} + 426880 p^{2} T^{10} + 79760 p^{3} T^{11} + 14204 p^{4} T^{12} + 1392 p^{5} T^{13} + 128 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \)
67 \( 1 - 16 T + 128 T^{2} - 304 T^{3} + 4388 T^{4} - 107696 T^{5} + 1207680 T^{6} - 4800272 T^{7} + 13154790 T^{8} - 4800272 p T^{9} + 1207680 p^{2} T^{10} - 107696 p^{3} T^{11} + 4388 p^{4} T^{12} - 304 p^{5} T^{13} + 128 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \)
71 \( 1 - 440 T^{2} + 90844 T^{4} - 11522952 T^{6} + 984512390 T^{8} - 11522952 p^{2} T^{10} + 90844 p^{4} T^{12} - 440 p^{6} T^{14} + p^{8} T^{16} \)
73 \( 1 - 328 T^{2} + 45404 T^{4} - 3734648 T^{6} + 259745542 T^{8} - 3734648 p^{2} T^{10} + 45404 p^{4} T^{12} - 328 p^{6} T^{14} + p^{8} T^{16} \)
79 \( ( 1 + 12 T + 148 T^{2} - 44 T^{3} + 794 T^{4} - 44 p T^{5} + 148 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( 1 - 40 T + 800 T^{2} - 11000 T^{3} + 122436 T^{4} - 1297720 T^{5} + 14460000 T^{6} - 161033000 T^{7} + 1597489574 T^{8} - 161033000 p T^{9} + 14460000 p^{2} T^{10} - 1297720 p^{3} T^{11} + 122436 p^{4} T^{12} - 11000 p^{5} T^{13} + 800 p^{6} T^{14} - 40 p^{7} T^{15} + p^{8} T^{16} \)
89 \( 1 - 248 T^{2} + 36316 T^{4} - 4626504 T^{6} + 476004998 T^{8} - 4626504 p^{2} T^{10} + 36316 p^{4} T^{12} - 248 p^{6} T^{14} + p^{8} T^{16} \)
97 \( ( 1 + 164 T^{2} + 768 T^{3} + 13510 T^{4} + 768 p T^{5} + 164 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.95366614377944373807127284371, −4.75341720893686343626649109457, −4.68620386841520595558449108363, −4.47292682848476598746833523159, −4.44246871174000371768462021508, −4.37057186045183257531550490965, −4.20965764239114577503413450834, −4.19431302167782812366478000999, −4.07705397765460023138055660320, −3.39846258282795762731619665786, −3.37946322201050372380329763652, −3.30688755352605506775610469418, −3.25959903538321491543264066094, −3.20774815438433283975246711094, −2.91289612241523218119961145656, −2.57687682205068795235215913770, −2.46333533971307305972152133665, −2.35500381271730125670881156796, −2.19916131676402039144251626707, −1.89020047741944756437274000865, −1.29448788262382590148833226916, −1.20603024807497834306164677842, −1.13638174546973384585626250038, −0.965250353819026023963088958118, −0.69554977315143057023148998299, 0.69554977315143057023148998299, 0.965250353819026023963088958118, 1.13638174546973384585626250038, 1.20603024807497834306164677842, 1.29448788262382590148833226916, 1.89020047741944756437274000865, 2.19916131676402039144251626707, 2.35500381271730125670881156796, 2.46333533971307305972152133665, 2.57687682205068795235215913770, 2.91289612241523218119961145656, 3.20774815438433283975246711094, 3.25959903538321491543264066094, 3.30688755352605506775610469418, 3.37946322201050372380329763652, 3.39846258282795762731619665786, 4.07705397765460023138055660320, 4.19431302167782812366478000999, 4.20965764239114577503413450834, 4.37057186045183257531550490965, 4.44246871174000371768462021508, 4.47292682848476598746833523159, 4.68620386841520595558449108363, 4.75341720893686343626649109457, 4.95366614377944373807127284371

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.