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Properties

Label 2-34e2-1156.47-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 − 0.638i)5-s + (−0.183 + 0.982i)8-s + (−0.673 + 0.739i)9-s + (1.11 + 0.156i)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 + (−1.07 + 0.359i)20-s + (0.100 + 0.258i)25-s + (1.29 − 0.368i)26-s + (−1.97 + 0.276i)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 − 0.638i)5-s + (−0.183 + 0.982i)8-s + (−0.673 + 0.739i)9-s + (1.11 + 0.156i)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 + (−1.07 + 0.359i)20-s + (0.100 + 0.258i)25-s + (1.29 − 0.368i)26-s + (−1.97 + 0.276i)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: $\chi_{1156} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :0),\ -0.952 - 0.304i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1426975447\)
\(L(\frac12)\) \(\approx\) \(0.1426975447\)
\(L(1)\) 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 \( 1 + (0.932 + 0.638i)T + (0.361 + 0.932i)T^{2} \)
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 \( 1 + (1.97 - 0.276i)T + (0.961 - 0.273i)T^{2} \)
31 \( 1 + (0.361 - 0.932i)T^{2} \)
37 \( 1 + (-0.434 - 1.84i)T + (-0.895 + 0.445i)T^{2} \)
41 \( 1 + (1.50 - 0.666i)T + (0.673 - 0.739i)T^{2} \)
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 \( 1 + (1.40 + 1.16i)T + (0.183 + 0.982i)T^{2} \)
67 \( 1 + (0.602 + 0.798i)T^{2} \)
71 \( 1 + (0.995 - 0.0922i)T^{2} \)
73 \( 1 + (-0.0449 + 0.0806i)T + (-0.526 - 0.850i)T^{2} \)
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 \( 1 + (0.0127 + 0.276i)T + (-0.995 + 0.0922i)T^{2} \)
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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

−10.25077546401393645731892744767, −9.416235934338003198022267327400, −8.554610758243142574383356851597, −7.889253409581182443847498647023, −7.49651301939415136454583086604, −6.28187582175593254467012273669, −5.30800379506326677227435542066, −4.59235911983533380291288334243, −3.09056059857157642694732162656, −1.75307865071205826874789685813, 0.15912076497982595295220522734, 2.23180505526925756832691151649, 3.22631927334571346795932229409, 3.96763526921074930867765402411, 5.41695576179585481013978684437, 6.68061339764338727569689978264, 7.40385858953891683155804329975, 7.83472155487255509444768600216, 9.110123530818608655604099123561, 9.417884327582685945202785220032

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