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Properties

Label 2-115-23.22-c6-0-39
Degree $2$
Conductor $115$
Sign $0.989 + 0.146i$
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  + 14.7·2-s + 22.8·3-s + 153.·4-s − 55.9i·5-s + 336.·6-s + 221. i·7-s + 1.31e3·8-s − 207.·9-s − 824. i·10-s − 1.82e3i·11-s + 3.49e3·12-s + 3.71e3·13-s + 3.25e3i·14-s − 1.27e3i·15-s + 9.58e3·16-s + 3.08e3i·17-s + ⋯
L(s)  = 1  + 1.84·2-s + 0.845·3-s + 2.39·4-s − 0.447i·5-s + 1.55·6-s + 0.644i·7-s + 2.57·8-s − 0.285·9-s − 0.824i·10-s − 1.37i·11-s + 2.02·12-s + 1.69·13-s + 1.18i·14-s − 0.378i·15-s + 2.34·16-s + 0.627i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.146i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.989 + 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.989 + 0.146i$
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: \((2,\ 115,\ (\ :3),\ 0.989 + 0.146i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(7.928311022\)
\(L(\frac12)\) \(\approx\) \(7.928311022\)
\(L(4)\) 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 \( 1 + (-1.77e3 + 1.20e4i)T \)
good2 \( 1 - 14.7T + 64T^{2} \)
3 \( 1 - 22.8T + 729T^{2} \)
7 \( 1 - 221. iT - 1.17e5T^{2} \)
11 \( 1 + 1.82e3iT - 1.77e6T^{2} \)
13 \( 1 - 3.71e3T + 4.82e6T^{2} \)
17 \( 1 - 3.08e3iT - 2.41e7T^{2} \)
19 \( 1 - 1.06e4iT - 4.70e7T^{2} \)
29 \( 1 + 2.16e4T + 5.94e8T^{2} \)
31 \( 1 + 3.97e4T + 8.87e8T^{2} \)
37 \( 1 + 2.71e4iT - 2.56e9T^{2} \)
41 \( 1 - 1.78e4T + 4.75e9T^{2} \)
43 \( 1 - 1.18e5iT - 6.32e9T^{2} \)
47 \( 1 + 1.38e5T + 1.07e10T^{2} \)
53 \( 1 + 1.70e5iT - 2.21e10T^{2} \)
59 \( 1 + 5.82e4T + 4.21e10T^{2} \)
61 \( 1 - 2.52e5iT - 5.15e10T^{2} \)
67 \( 1 - 2.27e5iT - 9.04e10T^{2} \)
71 \( 1 - 2.82e5T + 1.28e11T^{2} \)
73 \( 1 + 2.76e5T + 1.51e11T^{2} \)
79 \( 1 + 8.78e5iT - 2.43e11T^{2} \)
83 \( 1 + 3.77e5iT - 3.26e11T^{2} \)
89 \( 1 - 3.55e5iT - 4.96e11T^{2} \)
97 \( 1 + 1.88e5iT - 8.32e11T^{2} \)
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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

−12.84017468223878402210908326614, −11.61592516228052198258361113341, −10.83909790658478017786384404075, −8.832392111512086819141826403034, −8.057715910893626099823344896146, −6.08526703223913827715265684706, −5.68437320641400939755458836952, −3.88401762570707428828689050213, −3.21672244178337158356932092373, −1.75894809419727523880708744231, 1.92398663868661847523806895483, 3.18069054807439720522764513020, 4.03023277893731435079645461218, 5.36863598535390286582317754856, 6.76570005560423886873360739768, 7.54048800375860813539119140322, 9.262323207826473055834922797623, 10.87402805712180386055018145674, 11.51079510441565858049563240083, 12.92780443035697941377029454718

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