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Properties

Label 12-847e6-1.1-c1e6-0-3
Degree $12$
Conductor $3.692\times 10^{17}$
Sign $1$
Analytic cond. $95711.2$
Root an. cond. $2.60064$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s − 2·3-s + 4·4-s − 4·5-s + 8·6-s − 6·7-s + 4·8-s − 3·9-s + 16·10-s − 8·12-s − 4·13-s + 24·14-s + 8·15-s − 8·16-s − 22·17-s + 12·18-s − 6·19-s − 16·20-s + 12·21-s + 2·23-s − 8·24-s − 5·25-s + 16·26-s + 10·27-s − 24·28-s − 12·29-s − 32·30-s + ⋯
L(s)  = 1  − 2.82·2-s − 1.15·3-s + 2·4-s − 1.78·5-s + 3.26·6-s − 2.26·7-s + 1.41·8-s − 9-s + 5.05·10-s − 2.30·12-s − 1.10·13-s + 6.41·14-s + 2.06·15-s − 2·16-s − 5.33·17-s + 2.82·18-s − 1.37·19-s − 3.57·20-s + 2.61·21-s + 0.417·23-s − 1.63·24-s − 25-s + 3.13·26-s + 1.92·27-s − 4.53·28-s − 2.22·29-s − 5.84·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(7^{6} \cdot 11^{12}\)
Sign: $1$
Analytic conductor: \(95711.2\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 7^{6} \cdot 11^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( ( 1 + T )^{6} \)
11 \( 1 \)
good2 \( 1 + p^{2} T + 3 p^{2} T^{2} + 7 p^{2} T^{3} + 7 p^{3} T^{4} + 3 p^{5} T^{5} + 145 T^{6} + 3 p^{6} T^{7} + 7 p^{5} T^{8} + 7 p^{5} T^{9} + 3 p^{6} T^{10} + p^{7} T^{11} + p^{6} T^{12} \)
3 \( 1 + 2 T + 7 T^{2} + 10 T^{3} + 28 T^{4} + 4 p^{2} T^{5} + 100 T^{6} + 4 p^{3} T^{7} + 28 p^{2} T^{8} + 10 p^{3} T^{9} + 7 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 + 4 T + 21 T^{2} + 64 T^{3} + 194 T^{4} + 468 T^{5} + 1141 T^{6} + 468 p T^{7} + 194 p^{2} T^{8} + 64 p^{3} T^{9} + 21 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 4 T + 59 T^{2} + 188 T^{3} + 1511 T^{4} + 3984 T^{5} + 23754 T^{6} + 3984 p T^{7} + 1511 p^{2} T^{8} + 188 p^{3} T^{9} + 59 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 22 T + 284 T^{2} + 2536 T^{3} + 17485 T^{4} + 96354 T^{5} + 437477 T^{6} + 96354 p T^{7} + 17485 p^{2} T^{8} + 2536 p^{3} T^{9} + 284 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 6 T + 64 T^{2} + 428 T^{3} + 2540 T^{4} + 12238 T^{5} + 64622 T^{6} + 12238 p T^{7} + 2540 p^{2} T^{8} + 428 p^{3} T^{9} + 64 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 2 T + 53 T^{2} + 88 T^{3} + 1540 T^{4} + 3306 T^{5} + 47756 T^{6} + 3306 p T^{7} + 1540 p^{2} T^{8} + 88 p^{3} T^{9} + 53 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 12 T + 202 T^{2} + 1648 T^{3} + 15591 T^{4} + 93140 T^{5} + 613773 T^{6} + 93140 p T^{7} + 15591 p^{2} T^{8} + 1648 p^{3} T^{9} + 202 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 2 T + 68 T^{2} + 28 T^{3} + 3104 T^{4} + 2730 T^{5} + 128070 T^{6} + 2730 p T^{7} + 3104 p^{2} T^{8} + 28 p^{3} T^{9} + 68 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 14 T + 187 T^{2} - 1538 T^{3} + 323 p T^{4} - 73832 T^{5} + 476570 T^{6} - 73832 p T^{7} + 323 p^{3} T^{8} - 1538 p^{3} T^{9} + 187 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 26 T + 493 T^{2} + 6330 T^{3} + 66918 T^{4} + 558026 T^{5} + 3963849 T^{6} + 558026 p T^{7} + 66918 p^{2} T^{8} + 6330 p^{3} T^{9} + 493 p^{4} T^{10} + 26 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 4 T + 137 T^{2} - 608 T^{3} + 8804 T^{4} - 41340 T^{5} + 407244 T^{6} - 41340 p T^{7} + 8804 p^{2} T^{8} - 608 p^{3} T^{9} + 137 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 16 T + 255 T^{2} + 2338 T^{3} + 22256 T^{4} + 151482 T^{5} + 1182208 T^{6} + 151482 p T^{7} + 22256 p^{2} T^{8} + 2338 p^{3} T^{9} + 255 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 4 T + 238 T^{2} - 1038 T^{3} + 27039 T^{4} - 104266 T^{5} + 1829061 T^{6} - 104266 p T^{7} + 27039 p^{2} T^{8} - 1038 p^{3} T^{9} + 238 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 4 T + 67 T^{2} + 234 T^{3} + 3960 T^{4} + 16822 T^{5} + 97728 T^{6} + 16822 p T^{7} + 3960 p^{2} T^{8} + 234 p^{3} T^{9} + 67 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 8 T + 43 T^{2} - 374 T^{3} + 8672 T^{4} - 41834 T^{5} + 220388 T^{6} - 41834 p T^{7} + 8672 p^{2} T^{8} - 374 p^{3} T^{9} + 43 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 6 T + 241 T^{2} - 20 p T^{3} + 30488 T^{4} - 149518 T^{5} + 2474324 T^{6} - 149518 p T^{7} + 30488 p^{2} T^{8} - 20 p^{4} T^{9} + 241 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 22 T + 533 T^{2} - 7336 T^{3} + 101080 T^{4} - 997626 T^{5} + 9693428 T^{6} - 997626 p T^{7} + 101080 p^{2} T^{8} - 7336 p^{3} T^{9} + 533 p^{4} T^{10} - 22 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 14 T + 316 T^{2} + 3752 T^{3} + 49196 T^{4} + 458618 T^{5} + 4569698 T^{6} + 458618 p T^{7} + 49196 p^{2} T^{8} + 3752 p^{3} T^{9} + 316 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 28 T + 601 T^{2} - 8392 T^{3} + 100724 T^{4} - 981724 T^{5} + 9215300 T^{6} - 981724 p T^{7} + 100724 p^{2} T^{8} - 8392 p^{3} T^{9} + 601 p^{4} T^{10} - 28 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 22 T + 540 T^{2} + 6988 T^{3} + 99116 T^{4} + 932526 T^{5} + 10112698 T^{6} + 932526 p T^{7} + 99116 p^{2} T^{8} + 6988 p^{3} T^{9} + 540 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 280 T^{2} + 1400 T^{3} + 32241 T^{4} + 335650 T^{5} + 2775213 T^{6} + 335650 p T^{7} + 32241 p^{2} T^{8} + 1400 p^{3} T^{9} + 280 p^{4} T^{10} + p^{6} T^{12} \)
97 \( 1 + 4 T + 429 T^{2} + 1520 T^{3} + 87938 T^{4} + 265044 T^{5} + 110821 p T^{6} + 265044 p T^{7} + 87938 p^{2} T^{8} + 1520 p^{3} T^{9} + 429 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.93517308136166408588720967996, −5.91162918799560542352468053083, −5.51425236850674409658562045761, −5.45035039158392877488130066415, −5.39191367522529546264150136695, −5.22218825339016560749926027054, −4.84027746125310991138007646162, −4.60946958450508369006429540874, −4.52848967259081485390904781680, −4.42407652456914028532917430642, −4.22984543438485394805628710969, −4.07537894568469433886434490736, −4.05011689087491459350730501132, −3.53265249391648651910892017062, −3.48661506536455406332857560333, −3.46328327448619816863088209241, −3.28453108231070537346553490791, −2.70715440237354428203622246974, −2.68154648708726456744149963679, −2.35167463705777176244585519280, −2.19153157190981280822952148876, −2.10000486205875759830003682145, −1.93084857757062747794727626976, −1.14335703309485490666222457065, −1.00105339286632749263155777480, 0, 0, 0, 0, 0, 0, 1.00105339286632749263155777480, 1.14335703309485490666222457065, 1.93084857757062747794727626976, 2.10000486205875759830003682145, 2.19153157190981280822952148876, 2.35167463705777176244585519280, 2.68154648708726456744149963679, 2.70715440237354428203622246974, 3.28453108231070537346553490791, 3.46328327448619816863088209241, 3.48661506536455406332857560333, 3.53265249391648651910892017062, 4.05011689087491459350730501132, 4.07537894568469433886434490736, 4.22984543438485394805628710969, 4.42407652456914028532917430642, 4.52848967259081485390904781680, 4.60946958450508369006429540874, 4.84027746125310991138007646162, 5.22218825339016560749926027054, 5.39191367522529546264150136695, 5.45035039158392877488130066415, 5.51425236850674409658562045761, 5.91162918799560542352468053083, 5.93517308136166408588720967996

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

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