Properties

Label 2-34e2-1156.1123-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 + (0.726 − 0.961i)5-s + (−0.602 − 0.798i)8-s + (−0.850 − 0.526i)9-s + (−0.111 − 1.20i)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 + (−0.890 − 0.811i)20-s + (−0.123 − 0.435i)25-s + (1.67 + 0.312i)26-s + (−0.156 + 1.69i)29-s + (−0.850 + 0.526i)32-s + ⋯
L(s)  = 1  + (0.739 − 0.673i)2-s + (0.0922 − 0.995i)4-s + (0.726 − 0.961i)5-s + (−0.602 − 0.798i)8-s + (−0.850 − 0.526i)9-s + (−0.111 − 1.20i)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 + (−0.890 − 0.811i)20-s + (−0.123 − 0.435i)25-s + (1.67 + 0.312i)26-s + (−0.156 + 1.69i)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: $\chi_{1156} (1123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :0),\ -0.252 + 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.609232098\)
\(L(\frac12)\) \(\approx\) \(1.609232098\)
\(L(1)\) 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 \( 1 + (-0.726 + 0.961i)T + (-0.273 - 0.961i)T^{2} \)
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 \( 1 + (0.156 - 1.69i)T + (-0.982 - 0.183i)T^{2} \)
31 \( 1 + (0.273 - 0.961i)T^{2} \)
37 \( 1 + (-1.37 + 0.533i)T + (0.739 - 0.673i)T^{2} \)
41 \( 1 + (-0.329 + 1.15i)T + (-0.850 - 0.526i)T^{2} \)
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 \( 1 + (0.243 + 0.489i)T + (-0.602 + 0.798i)T^{2} \)
67 \( 1 + (-0.0922 - 0.995i)T^{2} \)
71 \( 1 + (-0.445 - 0.895i)T^{2} \)
73 \( 1 + (-1.67 - 0.312i)T + (0.932 + 0.361i)T^{2} \)
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 \( 1 + (0.156 + 0.0971i)T + (0.445 + 0.895i)T^{2} \)
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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

−9.533653798966588652708664903371, −9.153321119254892012765349119496, −8.572413710615375542941620030204, −6.88461374118163339951368401760, −6.15268344634926004126956112270, −5.41819275404912229925939063355, −4.51687446033115446550327774914, −3.63178619744009745386585385418, −2.34222585378623043777751110872, −1.28601908529817087469629756301, 2.40560666274826302324379782136, 3.03329303020964136862749841118, 4.21048304089933465495987291443, 5.39262543540979118941910256862, 6.10918117660060455511329197471, 6.52890189443163708210717118666, 7.84056742063706530218493863753, 8.223974109974248530842827269904, 9.312327733683215623345904970372, 10.40400539055841316805212408243

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