↑

Properties

Label 2-2156-44.3-c0-0-0
Degree $2$
Conductor $2156$
Sign $-0.286 - 0.958i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (−0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s + (0.809 + 0.587i)11-s + (−0.809 + 0.587i)16-s − 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 + (−0.809 − 0.587i)36-s + (−0.190 + 0.587i)37-s + 1.17i·43-s + (−0.309 + 0.951i)44-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (−0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s + (0.809 + 0.587i)11-s + (−0.809 + 0.587i)16-s − 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 + (−0.809 − 0.587i)36-s + (−0.190 + 0.587i)37-s + 1.17i·43-s + (−0.309 + 0.951i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.286 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.286 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2156\)    =    \(2^{2} \cdot 7^{2} \cdot 11\)
Sign: $-0.286 - 0.958i$
Analytic conductor: \(1.07598\)
Root analytic conductor: \(1.03729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2156} (883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2156,\ (\ :0),\ -0.286 - 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.731714765\)
\(L(\frac12)\) \(\approx\) \(1.731714765\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 - 0.587i)T \)
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 \( 1 + (0.309 - 0.951i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.395294917626435868040980214970, −8.443798418431548599369426063837, −7.87671405816392816643041156320, −7.09280914247039021216855352188, −6.22730742995106082540599559250, −5.60841165878992448041848328950, −4.70402838962780319898764472890, −3.94747622880843381280010349217, −2.94963164383529625575826051288, −1.93613290028870277772552978196, 0.963113067085820361643528446419, 2.34378197902843575970533307881, 3.30080513058424071049616693781, 3.95570889537423960554649426246, 5.00792384787480750087675018877, 5.80168470568820513954718102093, 6.49033458759685869512369331743, 7.18127077204117080544613769631, 8.675237519144662839495920612250, 8.912864791482909603938348477830

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