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Properties

Label 2-2156-44.15-c0-0-0
Degree $2$
Conductor $2156$
Sign $-0.822 - 0.568i$
Analytic cond. $1.07598$
Root an. cond. $1.03729$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.309 + 0.951i)2-s + (−0.809 − 0.587i)4-s + (0.809 − 0.587i)8-s + (−0.809 − 0.587i)9-s + (−0.809 + 0.587i)11-s + (0.309 + 0.951i)16-s + (0.809 − 0.587i)18-s + (−0.309 − 0.951i)22-s + 1.90i·23-s + (−0.309 + 0.951i)25-s + (0.5 + 1.53i)29-s − 32-s + (0.309 + 0.951i)36-s + (−0.190 − 0.587i)37-s + 1.17i·43-s + 0.999·44-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)2-s + (−0.809 − 0.587i)4-s + (0.809 − 0.587i)8-s + (−0.809 − 0.587i)9-s + (−0.809 + 0.587i)11-s + (0.309 + 0.951i)16-s + (0.809 − 0.587i)18-s + (−0.309 − 0.951i)22-s + 1.90i·23-s + (−0.309 + 0.951i)25-s + (0.5 + 1.53i)29-s − 32-s + (0.309 + 0.951i)36-s + (−0.190 − 0.587i)37-s + 1.17i·43-s + 0.999·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.822 - 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.822 - 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2156\)    =    \(2^{2} \cdot 7^{2} \cdot 11\)
Sign: $-0.822 - 0.568i$
Analytic conductor: \(1.07598\)
Root analytic conductor: \(1.03729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2156} (1863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2156,\ (\ :0),\ -0.822 - 0.568i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5978304981\)
\(L(\frac12)\) \(\approx\) \(0.5978304981\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 - 0.951i)T \)
7 \( 1 \)
11 \( 1 + (0.809 - 0.587i)T \)
good3 \( 1 + (0.809 + 0.587i)T^{2} \)
5 \( 1 + (0.309 - 0.951i)T^{2} \)
13 \( 1 + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (0.809 + 0.587i)T^{2} \)
23 \( 1 - 1.90iT - T^{2} \)
29 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (-0.809 - 0.587i)T^{2} \)
43 \( 1 - 1.17iT - T^{2} \)
47 \( 1 + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.309 - 0.951i)T^{2} \)
67 \( 1 - 1.17iT - T^{2} \)
71 \( 1 + (0.690 + 0.951i)T + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.690 + 0.951i)T + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (-0.309 + 0.951i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.309 + 0.951i)T^{2} \)
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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

−9.277803151725284698560150750564, −8.889559194824773682558100429175, −7.84202131972720242155577538013, −7.37984184573931236026362532338, −6.52730173290851273359128547822, −5.55000621923335633472208920160, −5.20392382048623533902776014333, −3.99432370900219462247714018816, −3.00846288659231015062940276482, −1.44882292241068811408419734826, 0.47548975821345062308886031626, 2.29961280087197328923065562649, 2.71655429696924485186212542077, 3.92023985262677330884460178430, 4.80219421442290289189964615938, 5.60628586288114440575739415966, 6.59790669053554305356205092444, 7.86346958204365734180921590450, 8.317397984434804676167563591192, 8.826687232056809798004631289991

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