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Properties

Label 2-2156-77.76-c1-0-8
Degree $2$
Conductor $2156$
Sign $-0.912 + 0.409i$
Analytic cond. $17.2157$
Root an. cond. $4.14918$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.42i·3-s + 4.39i·5-s + 0.971·9-s − 3.31·11-s − 6.26·15-s + 5.53·23-s − 14.3·25-s + 5.65i·27-s + 10.5i·31-s − 4.72i·33-s − 11.0·37-s + 4.27i·45-s − 1.53i·47-s − 13.6·53-s − 14.5i·55-s + ⋯
L(s)  = 1  + 0.822i·3-s + 1.96i·5-s + 0.323·9-s − 1.00·11-s − 1.61·15-s + 1.15·23-s − 2.87·25-s + 1.08i·27-s + 1.88i·31-s − 0.822i·33-s − 1.81·37-s + 0.636i·45-s − 0.224i·47-s − 1.87·53-s − 1.96i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 + 0.409i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.912 + 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2156\)    =    \(2^{2} \cdot 7^{2} \cdot 11\)
Sign: $-0.912 + 0.409i$
Analytic conductor: \(17.2157\)
Root analytic conductor: \(4.14918\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2156} (1077, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2156,\ (\ :1/2),\ -0.912 + 0.409i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.188016680\)
\(L(\frac12)\) \(\approx\) \(1.188016680\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
11 \( 1 + 3.31T \)
good3 \( 1 - 1.42iT - 3T^{2} \)
5 \( 1 - 4.39iT - 5T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 5.53T + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 10.5iT - 31T^{2} \)
37 \( 1 + 11.0T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 1.53iT - 47T^{2} \)
53 \( 1 + 13.6T + 53T^{2} \)
59 \( 1 + 6.50iT - 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 - 15.1T + 67T^{2} \)
71 \( 1 - 7.18T + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 18.6iT - 89T^{2} \)
97 \( 1 - 2.45iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.859576386739412678486477479202, −8.837874881192989378299005825980, −7.83151258230599582013039323054, −7.01892956867538722102506316175, −6.65362647681778788089479545140, −5.45483231736121069514667019081, −4.73636395512168883308009447202, −3.41244992757988012086022058414, −3.18622553897629413271825603246, −1.97312268424282059330444378045, 0.41529967648469548530180046778, 1.38285046046928803013701306375, 2.31944056862193226342085772927, 3.83657630627181075621461080148, 4.80483469100443591480604877290, 5.29835989844732907861451107269, 6.22439627395830796902627644273, 7.29232961127153043216384595852, 7.951033257525272854234880227837, 8.474460706952172372472152627707

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