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Properties

Label 2-34e2-1156.387-c0-0-0
Degree $2$
Conductor $1156$
Sign $0.252 - 0.967i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.673 + 0.739i)2-s + (−0.0922 + 0.995i)4-s + (0.273 − 0.0381i)5-s + (−0.798 + 0.602i)8-s + (0.526 − 0.850i)9-s + (0.212 + 0.176i)10-s + (0.634 + 0.840i)13-s + (−0.982 − 0.183i)16-s + (0.982 + 0.183i)17-s + (0.982 − 0.183i)18-s + (0.0127 + 0.276i)20-s + (−0.888 + 0.252i)25-s + (−0.193 + 1.03i)26-s + (−1.49 + 1.24i)29-s + (−0.526 − 0.850i)32-s + ⋯
L(s)  = 1  + (0.673 + 0.739i)2-s + (−0.0922 + 0.995i)4-s + (0.273 − 0.0381i)5-s + (−0.798 + 0.602i)8-s + (0.526 − 0.850i)9-s + (0.212 + 0.176i)10-s + (0.634 + 0.840i)13-s + (−0.982 − 0.183i)16-s + (0.982 + 0.183i)17-s + (0.982 − 0.183i)18-s + (0.0127 + 0.276i)20-s + (−0.888 + 0.252i)25-s + (−0.193 + 1.03i)26-s + (−1.49 + 1.24i)29-s + (−0.526 − 0.850i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $0.252 - 0.967i$
Analytic conductor: \(0.576919\)
Root analytic conductor: \(0.759551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (387, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :0),\ 0.252 - 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.559766706\)
\(L(\frac12)\) \(\approx\) \(1.559766706\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.673 - 0.739i)T \)
17 \( 1 + (-0.982 - 0.183i)T \)
good3 \( 1 + (-0.526 + 0.850i)T^{2} \)
5 \( 1 + (-0.273 + 0.0381i)T + (0.961 - 0.273i)T^{2} \)
7 \( 1 + (-0.895 + 0.445i)T^{2} \)
11 \( 1 + (0.183 - 0.982i)T^{2} \)
13 \( 1 + (-0.634 - 0.840i)T + (-0.273 + 0.961i)T^{2} \)
19 \( 1 + (0.0922 + 0.995i)T^{2} \)
23 \( 1 + (-0.895 + 0.445i)T^{2} \)
29 \( 1 + (1.49 - 1.24i)T + (0.183 - 0.982i)T^{2} \)
31 \( 1 + (0.961 + 0.273i)T^{2} \)
37 \( 1 + (0.0844 + 0.0373i)T + (0.673 + 0.739i)T^{2} \)
41 \( 1 + (0.963 + 1.73i)T + (-0.526 + 0.850i)T^{2} \)
43 \( 1 + (0.932 - 0.361i)T^{2} \)
47 \( 1 + (-0.445 + 0.895i)T^{2} \)
53 \( 1 + (-0.942 + 1.52i)T + (-0.445 - 0.895i)T^{2} \)
59 \( 1 + (-0.602 - 0.798i)T^{2} \)
61 \( 1 + (0.922 + 0.309i)T + (0.798 + 0.602i)T^{2} \)
67 \( 1 + (-0.0922 - 0.995i)T^{2} \)
71 \( 1 + (0.895 - 0.445i)T^{2} \)
73 \( 1 + (-0.258 - 0.377i)T + (-0.361 + 0.932i)T^{2} \)
79 \( 1 + (-0.995 + 0.0922i)T^{2} \)
83 \( 1 + (-0.850 + 0.526i)T^{2} \)
89 \( 1 + (0.811 - 1.07i)T + (-0.273 - 0.961i)T^{2} \)
97 \( 1 + (0.292 + 1.24i)T + (-0.895 + 0.445i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.988154669050282538170335179308, −9.165653174068065803757911085461, −8.527143083639041900747221883323, −7.38150465784551095232645386052, −6.85475754513347857751680255197, −5.90183399951639702510847379117, −5.24885707852280528747663096022, −3.93258416640864775753014458720, −3.51142476787304076989260824308, −1.82597936689942816334792907441, 1.40186473171508875261297463725, 2.53687738481256097204479522739, 3.60969562787134336485492772331, 4.53976112683079378107398550923, 5.56309007165810352403266766713, 6.05528466201515401170013284702, 7.38050849905121601436543541671, 8.130934734745400960921200229794, 9.345437509207036782451364177827, 10.05149224966775837753004992238

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