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Properties

Label 2-115-23.22-c6-0-27
Degree $2$
Conductor $115$
Sign $0.0589 + 0.998i$
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  + 7.47·2-s − 46.2·3-s − 8.19·4-s + 55.9i·5-s − 345.·6-s + 394. i·7-s − 539.·8-s + 1.40e3·9-s + 417. i·10-s + 1.81e3i·11-s + 378.·12-s − 1.82e3·13-s + 2.94e3i·14-s − 2.58e3i·15-s − 3.50e3·16-s − 3.45e3i·17-s + ⋯
L(s)  = 1  + 0.933·2-s − 1.71·3-s − 0.128·4-s + 0.447i·5-s − 1.59·6-s + 1.14i·7-s − 1.05·8-s + 1.92·9-s + 0.417i·10-s + 1.36i·11-s + 0.219·12-s − 0.832·13-s + 1.07i·14-s − 0.765i·15-s − 0.855·16-s − 0.702i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0589 + 0.998i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.0589 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.0589 + 0.998i$
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: \((2,\ 115,\ (\ :3),\ 0.0589 + 0.998i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.2914393687\)
\(L(\frac12)\) \(\approx\) \(0.2914393687\)
\(L(4)\) 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 \( 1 + (1.21e4 - 717. i)T \)
good2 \( 1 - 7.47T + 64T^{2} \)
3 \( 1 + 46.2T + 729T^{2} \)
7 \( 1 - 394. iT - 1.17e5T^{2} \)
11 \( 1 - 1.81e3iT - 1.77e6T^{2} \)
13 \( 1 + 1.82e3T + 4.82e6T^{2} \)
17 \( 1 + 3.45e3iT - 2.41e7T^{2} \)
19 \( 1 + 7.53e3iT - 4.70e7T^{2} \)
29 \( 1 - 4.04e4T + 5.94e8T^{2} \)
31 \( 1 + 1.56e4T + 8.87e8T^{2} \)
37 \( 1 + 3.76e4iT - 2.56e9T^{2} \)
41 \( 1 + 4.45e3T + 4.75e9T^{2} \)
43 \( 1 + 1.21e4iT - 6.32e9T^{2} \)
47 \( 1 - 2.81e4T + 1.07e10T^{2} \)
53 \( 1 - 1.05e4iT - 2.21e10T^{2} \)
59 \( 1 + 1.70e5T + 4.21e10T^{2} \)
61 \( 1 - 3.54e5iT - 5.15e10T^{2} \)
67 \( 1 + 4.04e5iT - 9.04e10T^{2} \)
71 \( 1 - 2.41e5T + 1.28e11T^{2} \)
73 \( 1 - 2.70e4T + 1.51e11T^{2} \)
79 \( 1 - 3.26e5iT - 2.43e11T^{2} \)
83 \( 1 + 2.11e5iT - 3.26e11T^{2} \)
89 \( 1 + 1.13e6iT - 4.96e11T^{2} \)
97 \( 1 + 1.50e6iT - 8.32e11T^{2} \)
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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

−12.13508222770182362683061141135, −11.65638320718783971591192644058, −10.25204646734032935716002864518, −9.251824784552010455397825441246, −7.17361144341515904371459740626, −6.16752140134064970747012682639, −5.16887034283992347588929819384, −4.51867417802863697901907761579, −2.46413764973886621517869119917, −0.11552027670537197075358938201, 0.928249164648075737730133478694, 3.76764207563417186717998482406, 4.71926700590464142253907732592, 5.72730083233274751841615221575, 6.51896380846431712670325896104, 8.150860380733637734645069666288, 9.925937422365848941061058003945, 10.78519343530170567392103655743, 11.91783234545581123715338337334, 12.50565410268530293874496635147

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