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Properties

Label 2-115-23.22-c6-0-19
Degree $2$
Conductor $115$
Sign $0.552 - 0.833i$
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  − 15.2·2-s + 41.9·3-s + 169.·4-s + 55.9i·5-s − 640.·6-s − 492. i·7-s − 1.60e3·8-s + 1.02e3·9-s − 853. i·10-s + 2.36e3i·11-s + 7.09e3·12-s − 416.·13-s + 7.52e3i·14-s + 2.34e3i·15-s + 1.37e4·16-s + 1.66e3i·17-s + ⋯
L(s)  = 1  − 1.90·2-s + 1.55·3-s + 2.64·4-s + 0.447i·5-s − 2.96·6-s − 1.43i·7-s − 3.14·8-s + 1.41·9-s − 0.853i·10-s + 1.77i·11-s + 4.10·12-s − 0.189·13-s + 2.74i·14-s + 0.694i·15-s + 3.35·16-s + 0.337i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.552 - 0.833i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.552 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.552 - 0.833i$
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: \((2,\ 115,\ (\ :3),\ 0.552 - 0.833i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.320581292\)
\(L(\frac12)\) \(\approx\) \(1.320581292\)
\(L(4)\) 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 \( 1 + (-1.01e4 - 6.71e3i)T \)
good2 \( 1 + 15.2T + 64T^{2} \)
3 \( 1 - 41.9T + 729T^{2} \)
7 \( 1 + 492. iT - 1.17e5T^{2} \)
11 \( 1 - 2.36e3iT - 1.77e6T^{2} \)
13 \( 1 + 416.T + 4.82e6T^{2} \)
17 \( 1 - 1.66e3iT - 2.41e7T^{2} \)
19 \( 1 - 7.40e3iT - 4.70e7T^{2} \)
29 \( 1 + 2.83e4T + 5.94e8T^{2} \)
31 \( 1 - 3.56e4T + 8.87e8T^{2} \)
37 \( 1 - 5.04e4iT - 2.56e9T^{2} \)
41 \( 1 - 1.93e4T + 4.75e9T^{2} \)
43 \( 1 + 1.86e4iT - 6.32e9T^{2} \)
47 \( 1 - 1.99e5T + 1.07e10T^{2} \)
53 \( 1 + 1.07e4iT - 2.21e10T^{2} \)
59 \( 1 - 9.95e4T + 4.21e10T^{2} \)
61 \( 1 + 1.61e5iT - 5.15e10T^{2} \)
67 \( 1 + 2.27e5iT - 9.04e10T^{2} \)
71 \( 1 - 2.20e5T + 1.28e11T^{2} \)
73 \( 1 - 1.30e5T + 1.51e11T^{2} \)
79 \( 1 - 9.05e5iT - 2.43e11T^{2} \)
83 \( 1 - 1.02e6iT - 3.26e11T^{2} \)
89 \( 1 - 4.58e5iT - 4.96e11T^{2} \)
97 \( 1 - 5.67e4iT - 8.32e11T^{2} \)
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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

−12.42127660671395188402889192166, −10.88199142343157579779779463427, −9.936514062122518624212037310498, −9.563932058397830841939779146485, −8.190379906432100691241633846775, −7.46998049935181700466923975090, −6.90634747857038228426466210500, −3.79766936052919175166144397284, −2.37120890173633005483982066513, −1.32429313370797612507518939023, 0.69473776312113422074075476857, 2.30038775641595628855641359014, 3.00186491659614471603722140806, 5.87454410586037365875294105193, 7.39619485520273591870010004540, 8.540860201659466665758400455229, 8.822921585990349779380378462520, 9.430778611263533326718136093919, 10.90477074577118636731986377257, 11.90099735256371559787406402321

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