↑

Properties

Label 2-1232-1232.741-c0-0-0
Degree $2$
Conductor $1232$
Sign $0.966 - 0.256i$
Analytic cond. $0.614848$
Root an. cond. $0.784122$
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  + (−0.587 − 0.809i)2-s + (−0.309 + 0.951i)4-s + (−0.951 + 0.309i)7-s + (0.951 − 0.309i)8-s + (−0.587 + 0.809i)9-s + (0.809 − 0.587i)11-s + (0.809 + 0.587i)14-s + (−0.809 − 0.587i)16-s + 18-s + (−0.951 − 0.309i)22-s + 1.90i·23-s + (0.951 + 0.309i)25-s − i·28-s + (1.76 + 0.896i)29-s + i·32-s + ⋯
L(s)  = 1  + (−0.587 − 0.809i)2-s + (−0.309 + 0.951i)4-s + (−0.951 + 0.309i)7-s + (0.951 − 0.309i)8-s + (−0.587 + 0.809i)9-s + (0.809 − 0.587i)11-s + (0.809 + 0.587i)14-s + (−0.809 − 0.587i)16-s + 18-s + (−0.951 − 0.309i)22-s + 1.90i·23-s + (0.951 + 0.309i)25-s − i·28-s + (1.76 + 0.896i)29-s + i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1232\)    =    \(2^{4} \cdot 7 \cdot 11\)
Sign: $0.966 - 0.256i$
Analytic conductor: \(0.614848\)
Root analytic conductor: \(0.784122\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1232} (741, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1232,\ (\ :0),\ 0.966 - 0.256i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6489117398\)
\(L(\frac12)\) \(\approx\) \(0.6489117398\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.587 + 0.809i)T \)
7 \( 1 + (0.951 - 0.309i)T \)
11 \( 1 + (-0.809 + 0.587i)T \)
good3 \( 1 + (0.587 - 0.809i)T^{2} \)
5 \( 1 + (-0.951 - 0.309i)T^{2} \)
13 \( 1 + (-0.951 + 0.309i)T^{2} \)
17 \( 1 + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.587 - 0.809i)T^{2} \)
23 \( 1 - 1.90iT - T^{2} \)
29 \( 1 + (-1.76 - 0.896i)T + (0.587 + 0.809i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.412 - 0.809i)T + (-0.587 - 0.809i)T^{2} \)
41 \( 1 + (-0.809 - 0.587i)T^{2} \)
43 \( 1 + (0.221 + 0.221i)T + iT^{2} \)
47 \( 1 + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (-0.309 - 1.95i)T + (-0.951 + 0.309i)T^{2} \)
59 \( 1 + (0.587 + 0.809i)T^{2} \)
61 \( 1 + (0.951 + 0.309i)T^{2} \)
67 \( 1 + (0.221 - 0.221i)T - iT^{2} \)
71 \( 1 + (0.951 + 1.30i)T + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (0.951 + 0.690i)T + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (0.951 + 0.309i)T^{2} \)
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

−10.02190765106153097263614331153, −8.985508222394240747796632257368, −8.756720395114852532110997391642, −7.65370052115270694404476893120, −6.79353521536317118168759590536, −5.76034763182165160204358508638, −4.67842571790251283856538110337, −3.37816600255881236383048990798, −2.87836008686890698794943599836, −1.40624807326241964942146145526, 0.76189047201252158478853776405, 2.59003425618363610365452595590, 3.94395889201690143792304765044, 4.84391607162369303681316926007, 6.19907445349351421707570578801, 6.51319406833055909848742847382, 7.21759418210700034722806115839, 8.545654979028674289088485736958, 8.820120175295947136700313967289, 9.910494402572442928689207562611

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