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Properties

Label 2-115-23.22-c6-0-30
Degree $2$
Conductor $115$
Sign $0.663 + 0.748i$
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  − 12.7·2-s + 22.2·3-s + 98.9·4-s + 55.9i·5-s − 284.·6-s + 111. i·7-s − 446.·8-s − 233.·9-s − 713. i·10-s − 1.30e3i·11-s + 2.20e3·12-s + 2.22e3·13-s − 1.42e3i·14-s + 1.24e3i·15-s − 634.·16-s − 6.99e3i·17-s + ⋯
L(s)  = 1  − 1.59·2-s + 0.824·3-s + 1.54·4-s + 0.447i·5-s − 1.31·6-s + 0.326i·7-s − 0.872·8-s − 0.320·9-s − 0.713i·10-s − 0.977i·11-s + 1.27·12-s + 1.01·13-s − 0.520i·14-s + 0.368i·15-s − 0.154·16-s − 1.42i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.663 + 0.748i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.663 + 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.663 + 0.748i$
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: \((2,\ 115,\ (\ :3),\ 0.663 + 0.748i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.9780439413\)
\(L(\frac12)\) \(\approx\) \(0.9780439413\)
\(L(4)\) 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 \( 1 + (9.10e3 - 8.06e3i)T \)
good2 \( 1 + 12.7T + 64T^{2} \)
3 \( 1 - 22.2T + 729T^{2} \)
7 \( 1 - 111. iT - 1.17e5T^{2} \)
11 \( 1 + 1.30e3iT - 1.77e6T^{2} \)
13 \( 1 - 2.22e3T + 4.82e6T^{2} \)
17 \( 1 + 6.99e3iT - 2.41e7T^{2} \)
19 \( 1 - 2.69e3iT - 4.70e7T^{2} \)
29 \( 1 - 3.50e4T + 5.94e8T^{2} \)
31 \( 1 + 2.55e4T + 8.87e8T^{2} \)
37 \( 1 - 8.97e3iT - 2.56e9T^{2} \)
41 \( 1 - 7.08e4T + 4.75e9T^{2} \)
43 \( 1 + 1.37e5iT - 6.32e9T^{2} \)
47 \( 1 - 3.89e4T + 1.07e10T^{2} \)
53 \( 1 + 1.75e5iT - 2.21e10T^{2} \)
59 \( 1 - 1.81e5T + 4.21e10T^{2} \)
61 \( 1 + 2.85e5iT - 5.15e10T^{2} \)
67 \( 1 - 1.82e5iT - 9.04e10T^{2} \)
71 \( 1 - 2.76e5T + 1.28e11T^{2} \)
73 \( 1 + 4.34e4T + 1.51e11T^{2} \)
79 \( 1 - 1.46e5iT - 2.43e11T^{2} \)
83 \( 1 + 4.46e5iT - 3.26e11T^{2} \)
89 \( 1 + 1.03e6iT - 4.96e11T^{2} \)
97 \( 1 + 7.81e5iT - 8.32e11T^{2} \)
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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

−11.74090192768030616441561360004, −10.99714159995497824953803731293, −9.860142507546828804126761390433, −8.863811812381218934619092377145, −8.315024515150830325509337978622, −7.22885902737774157346359183741, −5.86206802387978633101012096265, −3.39918759488457865386294561273, −2.20601419773174343963594208462, −0.57316716954930265539932243817, 1.09815106208082022835082869907, 2.32908965223388340870780035662, 4.14230484446075995183508699785, 6.25850352375500665374944031534, 7.64645249526675887572407247558, 8.424786281234189665455785065816, 9.111247896510815842269658299643, 10.17090963535535742730584308898, 11.04901704458813890682618355684, 12.43872719900088733467472153308

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