Properties

Label 2-34e2-1156.611-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  + (−0.739 + 0.673i)2-s + (0.0922 − 0.995i)4-s + (−1.27 − 0.961i)5-s + (0.602 + 0.798i)8-s + (0.850 + 0.526i)9-s + (1.58 − 0.147i)10-s + (−1.02 − 1.35i)13-s + (−0.982 − 0.183i)16-s + (−0.982 − 0.183i)17-s + (−0.982 + 0.183i)18-s + (−1.07 + 1.17i)20-s + (0.423 + 1.48i)25-s + (1.67 + 0.312i)26-s + (−1.04 − 0.0971i)29-s + (0.850 − 0.526i)32-s + ⋯
L(s)  = 1  + (−0.739 + 0.673i)2-s + (0.0922 − 0.995i)4-s + (−1.27 − 0.961i)5-s + (0.602 + 0.798i)8-s + (0.850 + 0.526i)9-s + (1.58 − 0.147i)10-s + (−1.02 − 1.35i)13-s + (−0.982 − 0.183i)16-s + (−0.982 − 0.183i)17-s + (−0.982 + 0.183i)18-s + (−1.07 + 1.17i)20-s + (0.423 + 1.48i)25-s + (1.67 + 0.312i)26-s + (−1.04 − 0.0971i)29-s + (0.850 − 0.526i)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: $\chi_{1156} (611, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :0),\ -0.252 + 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3265661118\)
\(L(\frac12)\) \(\approx\) \(0.3265661118\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.739 - 0.673i)T \)
17 \( 1 + (0.982 + 0.183i)T \)
good3 \( 1 + (-0.850 - 0.526i)T^{2} \)
5 \( 1 + (1.27 + 0.961i)T + (0.273 + 0.961i)T^{2} \)
7 \( 1 + (0.445 + 0.895i)T^{2} \)
11 \( 1 + (-0.982 - 0.183i)T^{2} \)
13 \( 1 + (1.02 + 1.35i)T + (-0.273 + 0.961i)T^{2} \)
19 \( 1 + (-0.0922 - 0.995i)T^{2} \)
23 \( 1 + (0.445 + 0.895i)T^{2} \)
29 \( 1 + (1.04 + 0.0971i)T + (0.982 + 0.183i)T^{2} \)
31 \( 1 + (-0.273 + 0.961i)T^{2} \)
37 \( 1 + (0.486 + 1.25i)T + (-0.739 + 0.673i)T^{2} \)
41 \( 1 + (1.53 + 0.436i)T + (0.850 + 0.526i)T^{2} \)
43 \( 1 + (-0.932 + 0.361i)T^{2} \)
47 \( 1 + (-0.445 + 0.895i)T^{2} \)
53 \( 1 + (0.757 + 0.469i)T + (0.445 + 0.895i)T^{2} \)
59 \( 1 + (0.602 + 0.798i)T^{2} \)
61 \( 1 + (-1.72 + 0.857i)T + (0.602 - 0.798i)T^{2} \)
67 \( 1 + (-0.0922 - 0.995i)T^{2} \)
71 \( 1 + (0.445 + 0.895i)T^{2} \)
73 \( 1 + (-0.193 + 1.03i)T + (-0.932 - 0.361i)T^{2} \)
79 \( 1 + (0.0922 + 0.995i)T^{2} \)
83 \( 1 + (0.850 - 0.526i)T^{2} \)
89 \( 1 + (-0.890 + 1.17i)T + (-0.273 - 0.961i)T^{2} \)
97 \( 1 + (1.04 - 1.69i)T + (-0.445 - 0.895i)T^{2} \)
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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

−9.607305750895616519938695028713, −8.759341784135998873997691933779, −7.996345463500919967373758999904, −7.53204577814291255973715747278, −6.77802117867212991139396934752, −5.26320703712977998300950031543, −4.90527774524980488591202342043, −3.78727712710552147652770600707, −2.01101349520912127598580525766, −0.36238845056206168124369861738, 1.82648247560790348177308102230, 3.03711286703193514377961882709, 3.97800891149683069888548978462, 4.57112949563601441202357055163, 6.80187249312404601932640324185, 6.87006980199327037082034129025, 7.77260556351829262159940876608, 8.669203121020271736110216016612, 9.558206645866012204680972232361, 10.18136259815685029684262720580

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