Properties

Label 12-1680e6-1.1-c1e6-0-2
Degree $12$
Conductor $2.248\times 10^{19}$
Sign $1$
Analytic cond. $5.82798\times 10^{6}$
Root an. cond. $3.66263$
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  + 2·5-s − 3·9-s − 12·11-s + 12·19-s + 25-s − 4·29-s − 4·31-s + 4·41-s − 6·45-s − 3·49-s − 24·55-s + 32·59-s − 12·61-s − 12·71-s + 24·79-s + 6·81-s − 28·89-s + 24·95-s + 36·99-s − 44·101-s + 20·109-s + 18·121-s + 8·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 0.894·5-s − 9-s − 3.61·11-s + 2.75·19-s + 1/5·25-s − 0.742·29-s − 0.718·31-s + 0.624·41-s − 0.894·45-s − 3/7·49-s − 3.23·55-s + 4.16·59-s − 1.53·61-s − 1.42·71-s + 2.70·79-s + 2/3·81-s − 2.96·89-s + 2.46·95-s + 3.61·99-s − 4.37·101-s + 1.91·109-s + 1.63·121-s + 0.715·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 3^{6} \cdot 5^{6} \cdot 7^{6}\)
Sign: $1$
Analytic conductor: \(5.82798\times 10^{6}\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1680} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 3^{6} \cdot 5^{6} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.078720197\)
\(L(\frac12)\) \(\approx\) \(1.078720197\)
\(L(\frac{3}{2})\) 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 \( ( 1 + T^{2} )^{3} \)
5 \( 1 - 2 T + 3 T^{2} - 12 T^{3} + 3 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
7 \( ( 1 + T^{2} )^{3} \)
good11 \( ( 1 + 2 T + p T^{2} )^{6} \)
13 \( 1 - 34 T^{2} + 359 T^{4} - 2172 T^{6} + 359 p^{2} T^{8} - 34 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 - 70 T^{2} + 2415 T^{4} - 51220 T^{6} + 2415 p^{2} T^{8} - 70 p^{4} T^{10} + p^{6} T^{12} \)
19 \( ( 1 - 6 T + 53 T^{2} - 188 T^{3} + 53 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
23 \( 1 - 106 T^{2} + 5183 T^{4} - 150348 T^{6} + 5183 p^{2} T^{8} - 106 p^{4} T^{10} + p^{6} T^{12} \)
29 \( ( 1 + 2 T + 35 T^{2} + 76 T^{3} + 35 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( ( 1 + 2 T + 41 T^{2} - 60 T^{3} + 41 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( 1 - 46 T^{2} + 1399 T^{4} - 74788 T^{6} + 1399 p^{2} T^{8} - 46 p^{4} T^{10} + p^{6} T^{12} \)
41 \( ( 1 - 2 T + 63 T^{2} + 36 T^{3} + 63 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( 1 + 46 T^{2} + 2839 T^{4} + 118948 T^{6} + 2839 p^{2} T^{8} + 46 p^{4} T^{10} + p^{6} T^{12} \)
47 \( 1 - 154 T^{2} + 12143 T^{4} - 652332 T^{6} + 12143 p^{2} T^{8} - 154 p^{4} T^{10} + p^{6} T^{12} \)
53 \( 1 - 146 T^{2} + 14103 T^{4} - 884828 T^{6} + 14103 p^{2} T^{8} - 146 p^{4} T^{10} + p^{6} T^{12} \)
59 \( ( 1 - 16 T + 113 T^{2} - 608 T^{3} + 113 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
61 \( ( 1 + 6 T + 131 T^{2} + 484 T^{3} + 131 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( 1 - 274 T^{2} + 36103 T^{4} - 2962972 T^{6} + 36103 p^{2} T^{8} - 274 p^{4} T^{10} + p^{6} T^{12} \)
71 \( ( 1 + 2 T + p T^{2} )^{6} \)
73 \( 1 - 298 T^{2} + 43775 T^{4} - 3982284 T^{6} + 43775 p^{2} T^{8} - 298 p^{4} T^{10} + p^{6} T^{12} \)
79 \( ( 1 - 12 T + 221 T^{2} - 1576 T^{3} + 221 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 - 306 T^{2} + 47783 T^{4} - 4793948 T^{6} + 47783 p^{2} T^{8} - 306 p^{4} T^{10} + p^{6} T^{12} \)
89 \( ( 1 + 14 T + 319 T^{2} + 2532 T^{3} + 319 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( 1 - 26 T^{2} + 8719 T^{4} + 446932 T^{6} + 8719 p^{2} T^{8} - 26 p^{4} T^{10} + p^{6} T^{12} \)
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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

−5.20286636159279683599135014054, −4.65112403565697059063266718931, −4.59239213455625234961375095261, −4.58201445019367074307619770349, −4.47314146685261746469627808608, −3.97583967422781439248899551662, −3.92452534324292257502031178352, −3.71544286381283675176396632105, −3.70523218506629136680591324629, −3.35450482545903181755857206240, −3.14715236085892919591777443794, −3.03039466229066130997813957209, −2.92114043076329080577023644199, −2.90088565925480393859685847045, −2.44626544083457065456892001260, −2.37751122945413633789497289089, −2.35608963455564249349736641983, −2.18634632881611606317491414288, −1.96028284945980152550128341701, −1.50625122641719413256357108992, −1.33895383511386401285699749202, −1.05444459159725150700998759633, −1.02981581222319332012335970657, −0.30448326162175073305230041462, −0.23563322294345529994145419479, 0.23563322294345529994145419479, 0.30448326162175073305230041462, 1.02981581222319332012335970657, 1.05444459159725150700998759633, 1.33895383511386401285699749202, 1.50625122641719413256357108992, 1.96028284945980152550128341701, 2.18634632881611606317491414288, 2.35608963455564249349736641983, 2.37751122945413633789497289089, 2.44626544083457065456892001260, 2.90088565925480393859685847045, 2.92114043076329080577023644199, 3.03039466229066130997813957209, 3.14715236085892919591777443794, 3.35450482545903181755857206240, 3.70523218506629136680591324629, 3.71544286381283675176396632105, 3.92452534324292257502031178352, 3.97583967422781439248899551662, 4.47314146685261746469627808608, 4.58201445019367074307619770349, 4.59239213455625234961375095261, 4.65112403565697059063266718931, 5.20286636159279683599135014054

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

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