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Properties

Label 2-115-23.22-c6-0-13
Degree $2$
Conductor $115$
Sign $0.823 + 0.568i$
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  − 12.5·2-s − 47.4·3-s + 94.6·4-s + 55.9i·5-s + 597.·6-s − 458. i·7-s − 386.·8-s + 1.52e3·9-s − 704. i·10-s + 570. i·11-s − 4.49e3·12-s − 1.73e3·13-s + 5.77e3i·14-s − 2.65e3i·15-s − 1.18e3·16-s − 2.30e3i·17-s + ⋯
L(s)  = 1  − 1.57·2-s − 1.75·3-s + 1.47·4-s + 0.447i·5-s + 2.76·6-s − 1.33i·7-s − 0.755·8-s + 2.08·9-s − 0.704i·10-s + 0.428i·11-s − 2.59·12-s − 0.789·13-s + 2.10i·14-s − 0.785i·15-s − 0.290·16-s − 0.469i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.823 + 0.568i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.823 + 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.823 + 0.568i$
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: \((2,\ 115,\ (\ :3),\ 0.823 + 0.568i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.2234829059\)
\(L(\frac12)\) \(\approx\) \(0.2234829059\)
\(L(4)\) 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 \( 1 + (6.91e3 - 1.00e4i)T \)
good2 \( 1 + 12.5T + 64T^{2} \)
3 \( 1 + 47.4T + 729T^{2} \)
7 \( 1 + 458. iT - 1.17e5T^{2} \)
11 \( 1 - 570. iT - 1.77e6T^{2} \)
13 \( 1 + 1.73e3T + 4.82e6T^{2} \)
17 \( 1 + 2.30e3iT - 2.41e7T^{2} \)
19 \( 1 - 6.68e3iT - 4.70e7T^{2} \)
29 \( 1 + 1.51e4T + 5.94e8T^{2} \)
31 \( 1 + 3.32e4T + 8.87e8T^{2} \)
37 \( 1 + 8.02e4iT - 2.56e9T^{2} \)
41 \( 1 - 5.37e4T + 4.75e9T^{2} \)
43 \( 1 - 1.13e5iT - 6.32e9T^{2} \)
47 \( 1 + 8.08e4T + 1.07e10T^{2} \)
53 \( 1 - 1.91e5iT - 2.21e10T^{2} \)
59 \( 1 - 2.39e5T + 4.21e10T^{2} \)
61 \( 1 + 3.55e4iT - 5.15e10T^{2} \)
67 \( 1 - 4.82e5iT - 9.04e10T^{2} \)
71 \( 1 + 3.62e5T + 1.28e11T^{2} \)
73 \( 1 + 5.20e5T + 1.51e11T^{2} \)
79 \( 1 - 2.01e5iT - 2.43e11T^{2} \)
83 \( 1 + 1.04e6iT - 3.26e11T^{2} \)
89 \( 1 + 1.32e5iT - 4.96e11T^{2} \)
97 \( 1 + 1.85e5iT - 8.32e11T^{2} \)
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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

−11.72672628569565312725922110430, −10.96979987532999435200288477196, −10.23809420928953066650209817816, −9.606093990539802584632659144446, −7.41370681954922648936296647919, −7.28416197965851836663342772962, −5.85754418538709924297755672079, −4.28237632557557190040388500891, −1.57368389827999354173321003482, −0.35229838711087284460541842591, 0.51061348975399355456409254632, 1.99597364731762315814726738510, 4.88729045180811182406790455637, 5.92475804496573282819811195758, 7.00363140261762112172030145691, 8.400227194538110365747545139427, 9.390992222118536104477767423474, 10.37640488807548386365202623645, 11.36033682544859522481021806706, 12.00049274723623337514088974843

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