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Properties

Label 2-34e2-1156.455-c0-0-0
Degree $2$
Conductor $1156$
Sign $0.952 - 0.304i$
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  + (0.895 + 0.445i)2-s + (0.602 + 0.798i)4-s + (−0.932 − 1.36i)5-s + (0.183 + 0.982i)8-s + (0.673 + 0.739i)9-s + (−0.227 − 1.63i)10-s + (1.32 − 0.247i)13-s + (−0.273 + 0.961i)16-s + (0.273 − 0.961i)17-s + (0.273 + 0.961i)18-s + (0.524 − 1.56i)20-s + (−0.622 + 1.60i)25-s + (1.29 + 0.368i)26-s + (0.0127 − 0.0914i)29-s + (−0.673 + 0.739i)32-s + ⋯
L(s)  = 1  + (0.895 + 0.445i)2-s + (0.602 + 0.798i)4-s + (−0.932 − 1.36i)5-s + (0.183 + 0.982i)8-s + (0.673 + 0.739i)9-s + (−0.227 − 1.63i)10-s + (1.32 − 0.247i)13-s + (−0.273 + 0.961i)16-s + (0.273 − 0.961i)17-s + (0.273 + 0.961i)18-s + (0.524 − 1.56i)20-s + (−0.622 + 1.60i)25-s + (1.29 + 0.368i)26-s + (0.0127 − 0.0914i)29-s + (−0.673 + 0.739i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $0.952 - 0.304i$
Analytic conductor: \(0.576919\)
Root analytic conductor: \(0.759551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (455, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :0),\ 0.952 - 0.304i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.646236614\)
\(L(\frac12)\) \(\approx\) \(1.646236614\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.895 - 0.445i)T \)
17 \( 1 + (-0.273 + 0.961i)T \)
good3 \( 1 + (-0.673 - 0.739i)T^{2} \)
5 \( 1 + (0.932 + 1.36i)T + (-0.361 + 0.932i)T^{2} \)
7 \( 1 + (0.995 + 0.0922i)T^{2} \)
11 \( 1 + (-0.961 - 0.273i)T^{2} \)
13 \( 1 + (-1.32 + 0.247i)T + (0.932 - 0.361i)T^{2} \)
19 \( 1 + (-0.602 + 0.798i)T^{2} \)
23 \( 1 + (0.995 + 0.0922i)T^{2} \)
29 \( 1 + (-0.0127 + 0.0914i)T + (-0.961 - 0.273i)T^{2} \)
31 \( 1 + (-0.361 - 0.932i)T^{2} \)
37 \( 1 + (0.618 + 0.145i)T + (0.895 + 0.445i)T^{2} \)
41 \( 1 + (0.456 - 1.03i)T + (-0.673 - 0.739i)T^{2} \)
43 \( 1 + (-0.850 + 0.526i)T^{2} \)
47 \( 1 + (-0.0922 - 0.995i)T^{2} \)
53 \( 1 + (1.34 + 1.47i)T + (-0.0922 + 0.995i)T^{2} \)
59 \( 1 + (-0.982 + 0.183i)T^{2} \)
61 \( 1 + (-0.516 - 0.621i)T + (-0.183 + 0.982i)T^{2} \)
67 \( 1 + (0.602 - 0.798i)T^{2} \)
71 \( 1 + (-0.995 - 0.0922i)T^{2} \)
73 \( 1 + (1.74 - 0.972i)T + (0.526 - 0.850i)T^{2} \)
79 \( 1 + (-0.798 - 0.602i)T^{2} \)
83 \( 1 + (0.739 + 0.673i)T^{2} \)
89 \( 1 + (1.75 + 0.328i)T + (0.932 + 0.361i)T^{2} \)
97 \( 1 + (-1.97 - 0.0914i)T + (0.995 + 0.0922i)T^{2} \)
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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

−10.05459564292355226428535025774, −8.853069664661217198807650120497, −8.212345972393903794750878246408, −7.63881718335639626743203321106, −6.70148763308032384453495113584, −5.52385151017163762015439824447, −4.83975744246636563633137723045, −4.15874793288860255845578195902, −3.21666784931427297528130653947, −1.51000763534474136452288407846, 1.60048085806193551603809729281, 3.14405107054416629226301574436, 3.66797133756073618766362552959, 4.38666743188274327178785696444, 5.88903041267553251071255457482, 6.54543116135613456719067293533, 7.16020132794589624250630156578, 8.155516269919666853379717139838, 9.342615021835964733759838589906, 10.43632423829110865465069173910

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