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Properties

Label 2-2156-7.2-c1-0-16
Degree $2$
Conductor $2156$
Sign $0.266 - 0.963i$
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  + (1.22 + 2.12i)3-s + (1 − 1.73i)5-s + (−1.49 + 2.59i)9-s + (0.5 + 0.866i)11-s + 0.449·13-s + 4.89·15-s + (−0.224 − 0.389i)17-s + (−2.44 + 4.24i)19-s + (0.449 − 0.778i)23-s + (0.500 + 0.866i)25-s + 2.89·29-s + (3.22 + 5.58i)31-s + (−1.22 + 2.12i)33-s + (−2 + 3.46i)37-s + (0.550 + 0.953i)39-s + ⋯
L(s)  = 1  + (0.707 + 1.22i)3-s + (0.447 − 0.774i)5-s + (−0.499 + 0.866i)9-s + (0.150 + 0.261i)11-s + 0.124·13-s + 1.26·15-s + (−0.0545 − 0.0944i)17-s + (−0.561 + 0.973i)19-s + (0.0937 − 0.162i)23-s + (0.100 + 0.173i)25-s + 0.538·29-s + (0.579 + 1.00i)31-s + (−0.213 + 0.369i)33-s + (−0.328 + 0.569i)37-s + (0.0881 + 0.152i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2156\)    =    \(2^{2} \cdot 7^{2} \cdot 11\)
Sign: $0.266 - 0.963i$
Analytic conductor: \(17.2157\)
Root analytic conductor: \(4.14918\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2156} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2156,\ (\ :1/2),\ 0.266 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.496847326\)
\(L(\frac12)\) \(\approx\) \(2.496847326\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
good3 \( 1 + (-1.22 - 2.12i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 - 0.449T + 13T^{2} \)
17 \( 1 + (0.224 + 0.389i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.44 - 4.24i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.449 + 0.778i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.89T + 29T^{2} \)
31 \( 1 + (-3.22 - 5.58i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2 - 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 9.34T + 41T^{2} \)
43 \( 1 - 2.89T + 43T^{2} \)
47 \( 1 + (3.22 - 5.58i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.89 + 8.48i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.67 - 6.36i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.67 + 4.63i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.34 - 12.7i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.8T + 71T^{2} \)
73 \( 1 + (-6.22 - 10.7i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.44 + 9.43i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 16.8T + 83T^{2} \)
89 \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.10T + 97T^{2} \)
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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

−9.261629653554903694908382435092, −8.608078216812273914101357097258, −8.090231378003636407730078911564, −6.88162424836201822471555448069, −5.91772151838115775552196068302, −4.99093346546856759359753151689, −4.41015229740952720612355163756, −3.57734281729963230746996854796, −2.59630038698535340076651467710, −1.32634255879246590084366556377, 0.872899299187999939596099229417, 2.21893125328564886253247714288, 2.64097941603382621854531427102, 3.74604109453515140714941932196, 4.95226098836850601719150364156, 6.33370276239633679899858706996, 6.42967416228549035664531229508, 7.43246032281911590151273875545, 7.980906517962795481908066141074, 8.847152912127815161194095970709

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