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Properties

Label 2-115-23.22-c6-0-11
Degree $2$
Conductor $115$
Sign $0.695 - 0.718i$
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  − 8.10·2-s − 4.25·3-s + 1.75·4-s − 55.9i·5-s + 34.4·6-s + 124. i·7-s + 504.·8-s − 710.·9-s + 453. i·10-s − 346. i·11-s − 7.46·12-s − 1.90e3·13-s − 1.01e3i·14-s + 237. i·15-s − 4.20e3·16-s − 6.96e3i·17-s + ⋯
L(s)  = 1  − 1.01·2-s − 0.157·3-s + 0.0274·4-s − 0.447i·5-s + 0.159·6-s + 0.363i·7-s + 0.985·8-s − 0.975·9-s + 0.453i·10-s − 0.260i·11-s − 0.00431·12-s − 0.866·13-s − 0.368i·14-s + 0.0704i·15-s − 1.02·16-s − 1.41i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.695 - 0.718i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.695 - 0.718i$
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: \((2,\ 115,\ (\ :3),\ 0.695 - 0.718i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.5009181088\)
\(L(\frac12)\) \(\approx\) \(0.5009181088\)
\(L(4)\) 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 \( 1 + (8.74e3 + 8.46e3i)T \)
good2 \( 1 + 8.10T + 64T^{2} \)
3 \( 1 + 4.25T + 729T^{2} \)
7 \( 1 - 124. iT - 1.17e5T^{2} \)
11 \( 1 + 346. iT - 1.77e6T^{2} \)
13 \( 1 + 1.90e3T + 4.82e6T^{2} \)
17 \( 1 + 6.96e3iT - 2.41e7T^{2} \)
19 \( 1 - 3.80e3iT - 4.70e7T^{2} \)
29 \( 1 + 1.91e4T + 5.94e8T^{2} \)
31 \( 1 - 2.31e4T + 8.87e8T^{2} \)
37 \( 1 - 3.83e4iT - 2.56e9T^{2} \)
41 \( 1 - 7.75e3T + 4.75e9T^{2} \)
43 \( 1 - 1.01e5iT - 6.32e9T^{2} \)
47 \( 1 - 1.93e5T + 1.07e10T^{2} \)
53 \( 1 - 2.77e5iT - 2.21e10T^{2} \)
59 \( 1 - 1.81e5T + 4.21e10T^{2} \)
61 \( 1 + 2.17e5iT - 5.15e10T^{2} \)
67 \( 1 - 4.55e5iT - 9.04e10T^{2} \)
71 \( 1 - 2.00e5T + 1.28e11T^{2} \)
73 \( 1 + 5.55e5T + 1.51e11T^{2} \)
79 \( 1 + 7.97e4iT - 2.43e11T^{2} \)
83 \( 1 - 1.47e5iT - 3.26e11T^{2} \)
89 \( 1 + 4.86e5iT - 4.96e11T^{2} \)
97 \( 1 + 1.66e6iT - 8.32e11T^{2} \)
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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

−12.29229969962458599290967442165, −11.42514123418909467835675725630, −10.16983658173575829415159072673, −9.244295707096614554569903054535, −8.433886285953477908269285762508, −7.41218206310465197740655930255, −5.75670777133141631980524840274, −4.57202292199668815578999268412, −2.53972757940844834009788544055, −0.75025086285265408142388611098, 0.36667098400655784425841420163, 2.12977590733226866906011357408, 3.98031442207600380267828668440, 5.55541757775997264339463537290, 7.05419621409274456166969802043, 8.035491074547250081066301255237, 9.076496230629009795373282696896, 10.15606176846462159795582749551, 10.88465960829538875078766809461, 12.04923051431030940822587364798

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