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Properties

Label 2-15-3.2-c10-0-8
Degree $2$
Conductor $15$
Sign $0.226 + 0.973i$
Analytic cond. $9.53035$
Root an. cond. $3.08712$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 17.3i·2-s + (−236. + 55.1i)3-s + 724.·4-s + 1.39e3i·5-s + (954. + 4.09e3i)6-s + 2.72e3·7-s − 3.02e4i·8-s + (5.29e4 − 2.60e4i)9-s + 2.42e4·10-s − 2.52e5i·11-s + (−1.71e5 + 3.99e4i)12-s + 502.·13-s − 4.72e4i·14-s + (−7.70e4 − 3.30e5i)15-s + 2.17e5·16-s − 2.15e6i·17-s + ⋯
L(s)  = 1  − 0.541i·2-s + (−0.973 + 0.226i)3-s + 0.707·4-s + 0.447i·5-s + (0.122 + 0.527i)6-s + 0.162·7-s − 0.923i·8-s + (0.897 − 0.441i)9-s + 0.242·10-s − 1.56i·11-s + (−0.688 + 0.160i)12-s + 0.00135·13-s − 0.0878i·14-s + (−0.101 − 0.435i)15-s + 0.207·16-s − 1.51i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.226 + 0.973i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.226 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.226 + 0.973i$
Analytic conductor: \(9.53035\)
Root analytic conductor: \(3.08712\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :5),\ 0.226 + 0.973i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.16924 - 0.928170i\)
\(L(\frac12)\) \(\approx\) \(1.16924 - 0.928170i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (236. - 55.1i)T \)
5 \( 1 - 1.39e3iT \)
good2 \( 1 + 17.3iT - 1.02e3T^{2} \)
7 \( 1 - 2.72e3T + 2.82e8T^{2} \)
11 \( 1 + 2.52e5iT - 2.59e10T^{2} \)
13 \( 1 - 502.T + 1.37e11T^{2} \)
17 \( 1 + 2.15e6iT - 2.01e12T^{2} \)
19 \( 1 - 3.98e6T + 6.13e12T^{2} \)
23 \( 1 - 6.75e6iT - 4.14e13T^{2} \)
29 \( 1 + 5.46e6iT - 4.20e14T^{2} \)
31 \( 1 - 1.48e7T + 8.19e14T^{2} \)
37 \( 1 - 1.82e7T + 4.80e15T^{2} \)
41 \( 1 + 1.19e8iT - 1.34e16T^{2} \)
43 \( 1 + 1.16e8T + 2.16e16T^{2} \)
47 \( 1 + 3.35e7iT - 5.25e16T^{2} \)
53 \( 1 - 1.31e8iT - 1.74e17T^{2} \)
59 \( 1 - 5.93e8iT - 5.11e17T^{2} \)
61 \( 1 - 1.26e9T + 7.13e17T^{2} \)
67 \( 1 + 2.41e9T + 1.82e18T^{2} \)
71 \( 1 - 2.75e9iT - 3.25e18T^{2} \)
73 \( 1 - 1.71e9T + 4.29e18T^{2} \)
79 \( 1 + 1.62e9T + 9.46e18T^{2} \)
83 \( 1 - 9.06e8iT - 1.55e19T^{2} \)
89 \( 1 - 5.65e9iT - 3.11e19T^{2} \)
97 \( 1 + 2.30e9T + 7.37e19T^{2} \)
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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

−16.42865125524421659874126712417, −15.67856074040073529599219643530, −13.68552353111437315442629449225, −11.74300790860806965684457198548, −11.25400557810086237277041644828, −9.823510780952486163708543231497, −7.19000341311980940885724632667, −5.64889246168673105754305606110, −3.22932653071270229673840059970, −0.876310177450797834369042757813, 1.62904178266446409792400513274, 4.91869405993100096460897098731, 6.45133419378445723319957049599, 7.78101659816827902406504723773, 10.17773945122685952001581249321, 11.69291554089658406805875337208, 12.74134324665077728007465562955, 14.84301437288199875546551220169, 16.03168957957414192809748166135, 17.08006241274885960349700740604

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