↑

Properties

Label 2-34e2-1156.591-c0-0-0
Degree $2$
Conductor $1156$
Sign $0.712 + 0.701i$
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  + (0.961 − 0.273i)2-s + (0.850 − 0.526i)4-s + (−0.0922 − 0.00426i)5-s + (0.673 − 0.739i)8-s + (−0.183 − 0.982i)9-s + (−0.0899 + 0.0211i)10-s + (0.271 + 0.247i)13-s + (0.445 − 0.895i)16-s + (−0.445 + 0.895i)17-s + (−0.445 − 0.895i)18-s + (−0.0806 + 0.0449i)20-s + (−0.987 − 0.0914i)25-s + (0.328 + 0.163i)26-s + (1.10 + 0.258i)29-s + (0.183 − 0.982i)32-s + ⋯
L(s)  = 1  + (0.961 − 0.273i)2-s + (0.850 − 0.526i)4-s + (−0.0922 − 0.00426i)5-s + (0.673 − 0.739i)8-s + (−0.183 − 0.982i)9-s + (−0.0899 + 0.0211i)10-s + (0.271 + 0.247i)13-s + (0.445 − 0.895i)16-s + (−0.445 + 0.895i)17-s + (−0.445 − 0.895i)18-s + (−0.0806 + 0.0449i)20-s + (−0.987 − 0.0914i)25-s + (0.328 + 0.163i)26-s + (1.10 + 0.258i)29-s + (0.183 − 0.982i)32-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $0.712 + 0.701i$
Analytic conductor: \(0.576919\)
Root analytic conductor: \(0.759551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (591, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :0),\ 0.712 + 0.701i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.824736187\)
\(L(\frac12)\) \(\approx\) \(1.824736187\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.961 + 0.273i)T \)
17 \( 1 + (0.445 - 0.895i)T \)
good3 \( 1 + (0.183 + 0.982i)T^{2} \)
5 \( 1 + (0.0922 + 0.00426i)T + (0.995 + 0.0922i)T^{2} \)
7 \( 1 + (0.361 + 0.932i)T^{2} \)
11 \( 1 + (0.895 + 0.445i)T^{2} \)
13 \( 1 + (-0.271 - 0.247i)T + (0.0922 + 0.995i)T^{2} \)
19 \( 1 + (-0.850 - 0.526i)T^{2} \)
23 \( 1 + (0.361 + 0.932i)T^{2} \)
29 \( 1 + (-1.10 - 0.258i)T + (0.895 + 0.445i)T^{2} \)
31 \( 1 + (0.995 - 0.0922i)T^{2} \)
37 \( 1 + (1.73 - 0.241i)T + (0.961 - 0.273i)T^{2} \)
41 \( 1 + (-1.53 - 1.27i)T + (0.183 + 0.982i)T^{2} \)
43 \( 1 + (-0.602 + 0.798i)T^{2} \)
47 \( 1 + (-0.932 - 0.361i)T^{2} \)
53 \( 1 + (-0.132 - 0.710i)T + (-0.932 + 0.361i)T^{2} \)
59 \( 1 + (0.739 + 0.673i)T^{2} \)
61 \( 1 + (0.621 - 1.40i)T + (-0.673 - 0.739i)T^{2} \)
67 \( 1 + (0.850 + 0.526i)T^{2} \)
71 \( 1 + (-0.361 - 0.932i)T^{2} \)
73 \( 1 + (1.56 - 0.524i)T + (0.798 - 0.602i)T^{2} \)
79 \( 1 + (0.526 - 0.850i)T^{2} \)
83 \( 1 + (-0.982 - 0.183i)T^{2} \)
89 \( 1 + (-1.42 + 1.29i)T + (0.0922 - 0.995i)T^{2} \)
97 \( 1 + (0.377 + 0.258i)T + (0.361 + 0.932i)T^{2} \)
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

−10.10492511742056217379071289876, −9.154839873202430637323479180613, −8.251614942948161318783346684859, −7.11735143051588560592017739437, −6.34517505138060714036492135907, −5.73151104280115858952386414944, −4.51566554931641439401072906122, −3.79812485024908953404262497780, −2.83432128654766148133224897709, −1.48239984381710691509668821097, 2.01798744635457746537230128243, 2.99596269212905659720011513023, 4.12335029344368065574523583465, 4.99518486248953683627490225597, 5.71490211232177865422749914318, 6.69596415908609816863529609809, 7.54659334746640246345760431533, 8.177667187822619878054588448172, 9.170551340215442807556776798116, 10.40096155304615449049003327415

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