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Properties

Label 2-34e2-1156.171-c0-0-0
Degree $2$
Conductor $1156$
Sign $0.887 + 0.460i$
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  + (−0.850 − 0.526i)2-s + (0.445 + 0.895i)4-s + (0.0170 + 0.183i)5-s + (0.0922 − 0.995i)8-s + (0.932 + 0.361i)9-s + (0.0822 − 0.165i)10-s + (0.172 − 1.85i)13-s + (−0.602 + 0.798i)16-s + (−0.602 + 0.798i)17-s + (−0.602 − 0.798i)18-s + (−0.156 + 0.0971i)20-s + (0.949 − 0.177i)25-s + (−1.12 + 1.48i)26-s + (0.831 + 1.66i)29-s + (0.932 − 0.361i)32-s + ⋯
L(s)  = 1  + (−0.850 − 0.526i)2-s + (0.445 + 0.895i)4-s + (0.0170 + 0.183i)5-s + (0.0922 − 0.995i)8-s + (0.932 + 0.361i)9-s + (0.0822 − 0.165i)10-s + (0.172 − 1.85i)13-s + (−0.602 + 0.798i)16-s + (−0.602 + 0.798i)17-s + (−0.602 − 0.798i)18-s + (−0.156 + 0.0971i)20-s + (0.949 − 0.177i)25-s + (−1.12 + 1.48i)26-s + (0.831 + 1.66i)29-s + (0.932 − 0.361i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $0.887 + 0.460i$
Analytic conductor: \(0.576919\)
Root analytic conductor: \(0.759551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :0),\ 0.887 + 0.460i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7723542455\)
\(L(\frac12)\) \(\approx\) \(0.7723542455\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.850 + 0.526i)T \)
17 \( 1 + (0.602 - 0.798i)T \)
good3 \( 1 + (-0.932 - 0.361i)T^{2} \)
5 \( 1 + (-0.0170 - 0.183i)T + (-0.982 + 0.183i)T^{2} \)
7 \( 1 + (-0.739 - 0.673i)T^{2} \)
11 \( 1 + (0.602 - 0.798i)T^{2} \)
13 \( 1 + (-0.172 + 1.85i)T + (-0.982 - 0.183i)T^{2} \)
19 \( 1 + (-0.445 + 0.895i)T^{2} \)
23 \( 1 + (-0.739 - 0.673i)T^{2} \)
29 \( 1 + (-0.831 - 1.66i)T + (-0.602 + 0.798i)T^{2} \)
31 \( 1 + (0.982 + 0.183i)T^{2} \)
37 \( 1 + (-0.465 + 1.63i)T + (-0.850 - 0.526i)T^{2} \)
41 \( 1 + (0.181 + 0.0339i)T + (0.932 + 0.361i)T^{2} \)
43 \( 1 + (0.273 - 0.961i)T^{2} \)
47 \( 1 + (-0.739 + 0.673i)T^{2} \)
53 \( 1 + (-1.37 - 0.533i)T + (0.739 + 0.673i)T^{2} \)
59 \( 1 + (-0.0922 + 0.995i)T^{2} \)
61 \( 1 + (1.45 + 1.32i)T + (0.0922 + 0.995i)T^{2} \)
67 \( 1 + (-0.445 + 0.895i)T^{2} \)
71 \( 1 + (-0.739 - 0.673i)T^{2} \)
73 \( 1 + (1.12 - 1.48i)T + (-0.273 - 0.961i)T^{2} \)
79 \( 1 + (-0.445 + 0.895i)T^{2} \)
83 \( 1 + (-0.932 + 0.361i)T^{2} \)
89 \( 1 + (0.156 + 1.69i)T + (-0.982 + 0.183i)T^{2} \)
97 \( 1 + (-0.831 - 0.322i)T + (0.739 + 0.673i)T^{2} \)
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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

−10.25015994314784665826251052627, −9.073859945789747553181036171205, −8.412976924467154628502987840882, −7.56939769053266226901422825568, −6.93236944455164994559250706208, −5.82128572419878965221904714830, −4.58152985324728172748177021756, −3.49192391556492926419752601929, −2.53128946298729961469836672404, −1.19409123053474964611097160587, 1.26359413014448654490991500921, 2.45907468546747568195689857256, 4.22189945014423448607195130546, 4.88647119347948135699681688565, 6.28966686511274309968037154781, 6.76476919813235285430692352468, 7.50242379020442123633830183159, 8.595712553922845983825962680516, 9.183188401365288943121163420154, 9.833966130512109272327170612822

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