↑

Properties

Label 2-34e2-1156.375-c0-0-0
Degree $2$
Conductor $1156$
Sign $-0.701 - 0.712i$
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  + (−0.273 + 0.961i)2-s + (−0.850 − 0.526i)4-s + (1.09 + 0.995i)5-s + (0.739 − 0.673i)8-s + (−0.982 − 0.183i)9-s + (−1.25 + 0.778i)10-s + (−1.45 + 1.32i)13-s + (0.445 + 0.895i)16-s + (0.445 + 0.895i)17-s + (0.445 − 0.895i)18-s + (−0.404 − 1.42i)20-s + (0.109 + 1.17i)25-s + (−0.876 − 1.75i)26-s + (1.67 + 1.03i)29-s + (−0.982 + 0.183i)32-s + ⋯
L(s)  = 1  + (−0.273 + 0.961i)2-s + (−0.850 − 0.526i)4-s + (1.09 + 0.995i)5-s + (0.739 − 0.673i)8-s + (−0.982 − 0.183i)9-s + (−1.25 + 0.778i)10-s + (−1.45 + 1.32i)13-s + (0.445 + 0.895i)16-s + (0.445 + 0.895i)17-s + (0.445 − 0.895i)18-s + (−0.404 − 1.42i)20-s + (0.109 + 1.17i)25-s + (−0.876 − 1.75i)26-s + (1.67 + 1.03i)29-s + (−0.982 + 0.183i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $-0.701 - 0.712i$
Analytic conductor: \(0.576919\)
Root analytic conductor: \(0.759551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (375, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :0),\ -0.701 - 0.712i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8563595889\)
\(L(\frac12)\) \(\approx\) \(0.8563595889\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.273 - 0.961i)T \)
17 \( 1 + (-0.445 - 0.895i)T \)
good3 \( 1 + (0.982 + 0.183i)T^{2} \)
5 \( 1 + (-1.09 - 0.995i)T + (0.0922 + 0.995i)T^{2} \)
7 \( 1 + (-0.932 - 0.361i)T^{2} \)
11 \( 1 + (-0.445 - 0.895i)T^{2} \)
13 \( 1 + (1.45 - 1.32i)T + (0.0922 - 0.995i)T^{2} \)
19 \( 1 + (0.850 - 0.526i)T^{2} \)
23 \( 1 + (-0.932 - 0.361i)T^{2} \)
29 \( 1 + (-1.67 - 1.03i)T + (0.445 + 0.895i)T^{2} \)
31 \( 1 + (-0.0922 + 0.995i)T^{2} \)
37 \( 1 + (-0.329 - 0.436i)T + (-0.273 + 0.961i)T^{2} \)
41 \( 1 + (-0.136 + 1.47i)T + (-0.982 - 0.183i)T^{2} \)
43 \( 1 + (0.602 + 0.798i)T^{2} \)
47 \( 1 + (-0.932 + 0.361i)T^{2} \)
53 \( 1 + (1.83 + 0.342i)T + (0.932 + 0.361i)T^{2} \)
59 \( 1 + (-0.739 + 0.673i)T^{2} \)
61 \( 1 + (-0.172 - 0.0666i)T + (0.739 + 0.673i)T^{2} \)
67 \( 1 + (0.850 - 0.526i)T^{2} \)
71 \( 1 + (-0.932 - 0.361i)T^{2} \)
73 \( 1 + (0.876 + 1.75i)T + (-0.602 + 0.798i)T^{2} \)
79 \( 1 + (0.850 - 0.526i)T^{2} \)
83 \( 1 + (0.982 - 0.183i)T^{2} \)
89 \( 1 + (0.404 + 0.368i)T + (0.0922 + 0.995i)T^{2} \)
97 \( 1 + (-1.67 - 0.312i)T + (0.932 + 0.361i)T^{2} \)
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

−10.17498748189973755384957956010, −9.364369583629410239800649844667, −8.762883869714964361025263145182, −7.68733132265928789970097978944, −6.79451942785087842439763683375, −6.30961952265729104644506766670, −5.48672023804609392908304892759, −4.54236210323809808237790988735, −3.09388174612945160688824368725, −1.92383587300622588798351419297, 0.850730756124675275330105089148, 2.39415563935079843228643075734, 2.96715539179495410502403970997, 4.68181236645547792697953891712, 5.17272433818413914177593384793, 5.99957303580887622659680052670, 7.60392472927279573344576576042, 8.267931069885464755022162971854, 9.078436745206874386359748733090, 9.865036056923953572359616650425

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