↑

Properties

Label 2-34e2-68.43-c0-0-0
Degree $2$
Conductor $1156$
Sign $-0.682 - 0.730i$
Analytic cond. $0.576919$
Root an. cond. $0.759551$
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  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (−0.541 + 1.30i)5-s + (0.707 + 0.707i)8-s + (0.707 + 0.707i)9-s + (−0.541 − 1.30i)10-s − 1.00·16-s − 1.00·18-s + (1.30 + 0.541i)20-s + (−0.707 − 0.707i)25-s + (−0.541 + 1.30i)29-s + (0.707 − 0.707i)32-s + (0.707 − 0.707i)36-s + (−1.30 − 0.541i)37-s + (−1.30 + 0.541i)40-s + (0.541 + 1.30i)41-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (−0.541 + 1.30i)5-s + (0.707 + 0.707i)8-s + (0.707 + 0.707i)9-s + (−0.541 − 1.30i)10-s − 1.00·16-s − 1.00·18-s + (1.30 + 0.541i)20-s + (−0.707 − 0.707i)25-s + (−0.541 + 1.30i)29-s + (0.707 − 0.707i)32-s + (0.707 − 0.707i)36-s + (−1.30 − 0.541i)37-s + (−1.30 + 0.541i)40-s + (0.541 + 1.30i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.682 - 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.682 - 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $-0.682 - 0.730i$
Analytic conductor: \(0.576919\)
Root analytic conductor: \(0.759551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :0),\ -0.682 - 0.730i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6298527204\)
\(L(\frac12)\) \(\approx\) \(0.6298527204\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 \)
good3 \( 1 + (-0.707 - 0.707i)T^{2} \)
5 \( 1 + (0.541 - 1.30i)T + (-0.707 - 0.707i)T^{2} \)
7 \( 1 + (0.707 - 0.707i)T^{2} \)
11 \( 1 + (-0.707 + 0.707i)T^{2} \)
13 \( 1 - T^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 + (-0.707 + 0.707i)T^{2} \)
29 \( 1 + (0.541 - 1.30i)T + (-0.707 - 0.707i)T^{2} \)
31 \( 1 + (-0.707 - 0.707i)T^{2} \)
37 \( 1 + (1.30 + 0.541i)T + (0.707 + 0.707i)T^{2} \)
41 \( 1 + (-0.541 - 1.30i)T + (-0.707 + 0.707i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 + (-0.541 - 1.30i)T + (-0.707 + 0.707i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.707 - 0.707i)T^{2} \)
73 \( 1 + (-0.541 + 1.30i)T + (-0.707 - 0.707i)T^{2} \)
79 \( 1 + (-0.707 + 0.707i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.541 + 1.30i)T + (-0.707 - 0.707i)T^{2} \)
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

−10.33247392518356803211830796657, −9.497638579998251220139941357241, −8.508163179093383524084798205624, −7.60360792509367313076915986091, −7.17327618002241218645316118322, −6.45013250596938970202618493246, −5.38084626813150845815464027475, −4.35029249334929860157248360240, −3.08203415893466478513500146656, −1.76431244579556931700635892351, 0.74323389126179422601511388332, 1.96221347260148237199134226628, 3.55934054757446103978096363008, 4.20793446163963183863860167720, 5.16832130570221569520932717966, 6.57536472882265877275954721629, 7.51719854346146224919590754450, 8.269857636964103829336318063498, 8.960521620835851013857347758429, 9.604641777032126863168522650357

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