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Properties

Label 2-115-23.22-c6-0-45
Degree $2$
Conductor $115$
Sign $0.989 + 0.141i$
Analytic cond. $26.4562$
Root an. cond. $5.14356$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 13.6·2-s + 46.5·3-s + 122.·4-s − 55.9i·5-s + 636.·6-s − 127. i·7-s + 801.·8-s + 1.44e3·9-s − 763. i·10-s + 2.51e3i·11-s + 5.71e3·12-s − 3.10e3·13-s − 1.74e3i·14-s − 2.60e3i·15-s + 3.09e3·16-s − 7.08e3i·17-s + ⋯
L(s)  = 1  + 1.70·2-s + 1.72·3-s + 1.91·4-s − 0.447i·5-s + 2.94·6-s − 0.372i·7-s + 1.56·8-s + 1.97·9-s − 0.763i·10-s + 1.89i·11-s + 3.30·12-s − 1.41·13-s − 0.636i·14-s − 0.771i·15-s + 0.755·16-s − 1.44i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.141i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.989 + 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.989 + 0.141i$
Analytic conductor: \(26.4562\)
Root analytic conductor: \(5.14356\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :3),\ 0.989 + 0.141i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(8.810937100\)
\(L(\frac12)\) \(\approx\) \(8.810937100\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + 55.9iT \)
23 \( 1 + (-1.71e3 + 1.20e4i)T \)
good2 \( 1 - 13.6T + 64T^{2} \)
3 \( 1 - 46.5T + 729T^{2} \)
7 \( 1 + 127. iT - 1.17e5T^{2} \)
11 \( 1 - 2.51e3iT - 1.77e6T^{2} \)
13 \( 1 + 3.10e3T + 4.82e6T^{2} \)
17 \( 1 + 7.08e3iT - 2.41e7T^{2} \)
19 \( 1 - 4.06e3iT - 4.70e7T^{2} \)
29 \( 1 - 3.05e3T + 5.94e8T^{2} \)
31 \( 1 - 1.39e3T + 8.87e8T^{2} \)
37 \( 1 + 1.37e4iT - 2.56e9T^{2} \)
41 \( 1 + 1.05e5T + 4.75e9T^{2} \)
43 \( 1 - 8.28e4iT - 6.32e9T^{2} \)
47 \( 1 - 1.25e5T + 1.07e10T^{2} \)
53 \( 1 - 1.84e5iT - 2.21e10T^{2} \)
59 \( 1 + 2.35e5T + 4.21e10T^{2} \)
61 \( 1 - 4.01e4iT - 5.15e10T^{2} \)
67 \( 1 + 7.09e3iT - 9.04e10T^{2} \)
71 \( 1 - 2.70e4T + 1.28e11T^{2} \)
73 \( 1 - 6.18e5T + 1.51e11T^{2} \)
79 \( 1 - 6.77e4iT - 2.43e11T^{2} \)
83 \( 1 + 9.18e5iT - 3.26e11T^{2} \)
89 \( 1 - 4.14e5iT - 4.96e11T^{2} \)
97 \( 1 + 3.65e5iT - 8.32e11T^{2} \)
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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

−12.56206625719224063612078672999, −12.15693941874268594866220592133, −10.13762011755109330062199902365, −9.255730476941128140702430987577, −7.59389770283718739834672741546, −7.02005593714588871451592637145, −4.86309457315618557039127880450, −4.29187787760906397846590206701, −2.84949620933428859668927523995, −2.02182334101641847051583175420, 2.14390074601316045057357251412, 3.09990240992374271922133410248, 3.81106333252050447884668267394, 5.39578044324322862718185393893, 6.72560918885305537730664052675, 7.955693034218104326115931143851, 9.038879084525967855815187765195, 10.52080044963134736457112690117, 11.78732052525214075595832198907, 12.89867898995800895999110573942

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