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Properties

Label 2-115-23.22-c6-0-12
Degree $2$
Conductor $115$
Sign $0.912 - 0.408i$
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  + 2.39·2-s − 20.5·3-s − 58.2·4-s + 55.9i·5-s − 49.2·6-s − 343. i·7-s − 292.·8-s − 306.·9-s + 133. i·10-s − 71.3i·11-s + 1.19e3·12-s − 3.43e3·13-s − 822. i·14-s − 1.14e3i·15-s + 3.02e3·16-s − 5.91e3i·17-s + ⋯
L(s)  = 1  + 0.299·2-s − 0.761·3-s − 0.910·4-s + 0.447i·5-s − 0.228·6-s − 1.00i·7-s − 0.572·8-s − 0.419·9-s + 0.133i·10-s − 0.0536i·11-s + 0.693·12-s − 1.56·13-s − 0.299i·14-s − 0.340i·15-s + 0.738·16-s − 1.20i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 - 0.408i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.912 - 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.912 - 0.408i$
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: \((2,\ 115,\ (\ :3),\ 0.912 - 0.408i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.7698792724\)
\(L(\frac12)\) \(\approx\) \(0.7698792724\)
\(L(4)\) 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 \( 1 + (-4.96e3 - 1.11e4i)T \)
good2 \( 1 - 2.39T + 64T^{2} \)
3 \( 1 + 20.5T + 729T^{2} \)
7 \( 1 + 343. iT - 1.17e5T^{2} \)
11 \( 1 + 71.3iT - 1.77e6T^{2} \)
13 \( 1 + 3.43e3T + 4.82e6T^{2} \)
17 \( 1 + 5.91e3iT - 2.41e7T^{2} \)
19 \( 1 - 7.28e3iT - 4.70e7T^{2} \)
29 \( 1 - 2.38e3T + 5.94e8T^{2} \)
31 \( 1 - 4.95e4T + 8.87e8T^{2} \)
37 \( 1 - 4.58e4iT - 2.56e9T^{2} \)
41 \( 1 - 2.44e3T + 4.75e9T^{2} \)
43 \( 1 + 8.57e4iT - 6.32e9T^{2} \)
47 \( 1 + 1.11e5T + 1.07e10T^{2} \)
53 \( 1 - 1.86e5iT - 2.21e10T^{2} \)
59 \( 1 - 1.05e5T + 4.21e10T^{2} \)
61 \( 1 + 9.41e4iT - 5.15e10T^{2} \)
67 \( 1 + 3.39e5iT - 9.04e10T^{2} \)
71 \( 1 + 1.01e5T + 1.28e11T^{2} \)
73 \( 1 - 3.40e5T + 1.51e11T^{2} \)
79 \( 1 - 6.67e5iT - 2.43e11T^{2} \)
83 \( 1 - 5.67e5iT - 3.26e11T^{2} \)
89 \( 1 + 4.91e5iT - 4.96e11T^{2} \)
97 \( 1 + 7.10e5iT - 8.32e11T^{2} \)
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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

−12.35676606047727644903437665471, −11.60784981340315390340319350438, −10.31302921235659558301949259190, −9.582028008391846691829085766712, −7.992628485395175784029129389410, −6.83939921415782155287573116974, −5.45048265035296910662521176222, −4.54435927334685579292894207052, −3.07020719255016405774830209479, −0.65295630662119423657180404643, 0.46166541323541107803674669196, 2.65889788933104629213146297464, 4.59797813091187448482941227325, 5.28732565823378661627901385025, 6.38122846980163989521041847246, 8.242540694520118062427871852803, 9.075323992430948287435817030012, 10.18065103872535130398507904155, 11.63297260011291894476645004028, 12.38425453052853942761696154791

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