Properties

Label 2-115-23.22-c6-0-2
Degree $2$
Conductor $115$
Sign $-0.727 + 0.686i$
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  − 15.5·2-s − 20.6·3-s + 178.·4-s + 55.9i·5-s + 321.·6-s + 270. i·7-s − 1.77e3·8-s − 301.·9-s − 869. i·10-s − 1.14e3i·11-s − 3.68e3·12-s − 409.·13-s − 4.20e3i·14-s − 1.15e3i·15-s + 1.62e4·16-s + 9.72e3i·17-s + ⋯
L(s)  = 1  − 1.94·2-s − 0.765·3-s + 2.78·4-s + 0.447i·5-s + 1.48·6-s + 0.787i·7-s − 3.47·8-s − 0.413·9-s − 0.869i·10-s − 0.860i·11-s − 2.13·12-s − 0.186·13-s − 1.53i·14-s − 0.342i·15-s + 3.96·16-s + 1.98i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.727 + 0.686i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.727 + 0.686i$
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: \((2,\ 115,\ (\ :3),\ -0.727 + 0.686i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.1004629158\)
\(L(\frac12)\) \(\approx\) \(0.1004629158\)
\(L(4)\) 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 \( 1 + (8.35e3 + 8.84e3i)T \)
good2 \( 1 + 15.5T + 64T^{2} \)
3 \( 1 + 20.6T + 729T^{2} \)
7 \( 1 - 270. iT - 1.17e5T^{2} \)
11 \( 1 + 1.14e3iT - 1.77e6T^{2} \)
13 \( 1 + 409.T + 4.82e6T^{2} \)
17 \( 1 - 9.72e3iT - 2.41e7T^{2} \)
19 \( 1 - 1.11e4iT - 4.70e7T^{2} \)
29 \( 1 - 3.28e4T + 5.94e8T^{2} \)
31 \( 1 - 2.39e4T + 8.87e8T^{2} \)
37 \( 1 - 4.63e4iT - 2.56e9T^{2} \)
41 \( 1 + 1.18e5T + 4.75e9T^{2} \)
43 \( 1 - 1.05e5iT - 6.32e9T^{2} \)
47 \( 1 + 1.38e4T + 1.07e10T^{2} \)
53 \( 1 - 5.96e4iT - 2.21e10T^{2} \)
59 \( 1 + 1.00e5T + 4.21e10T^{2} \)
61 \( 1 - 1.54e4iT - 5.15e10T^{2} \)
67 \( 1 + 2.52e5iT - 9.04e10T^{2} \)
71 \( 1 + 1.21e5T + 1.28e11T^{2} \)
73 \( 1 + 4.15e5T + 1.51e11T^{2} \)
79 \( 1 - 5.66e4iT - 2.43e11T^{2} \)
83 \( 1 + 8.17e5iT - 3.26e11T^{2} \)
89 \( 1 + 1.85e5iT - 4.96e11T^{2} \)
97 \( 1 + 9.30e5iT - 8.32e11T^{2} \)
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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

−12.25583492379995678042257564620, −11.73013755830666471020302768903, −10.58605023374526549177938686582, −10.13668526668522014654121916886, −8.495780098671375314738833592308, −8.191947633557381648626967208680, −6.29400504924848905728026530982, −6.04949871028081835084445787439, −2.98089333685857423981745815630, −1.51322207675588541702670560506, 0.092839668507830973797097111969, 0.885688137148275341112420751600, 2.58132334882580066987805955514, 5.08922389429635860149757335870, 6.73170183468097083657498817029, 7.37071842271925853626937164155, 8.683846578039929545775527252029, 9.642119509650996314571237198959, 10.47594631134688951350298430257, 11.56925644412940078488180414696

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