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Properties

Label 2-15-15.2-c11-0-16
Degree $2$
Conductor $15$
Sign $-0.937 - 0.347i$
Analytic cond. $11.5251$
Root an. cond. $3.39487$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−61.8 + 61.8i)2-s + (3.69 − 420. i)3-s − 5.60e3i·4-s + (−4.34e3 − 5.47e3i)5-s + (2.58e4 + 2.62e4i)6-s + (−1.17e4 − 1.17e4i)7-s + (2.20e5 + 2.20e5i)8-s + (−1.77e5 − 3.11e3i)9-s + (6.07e5 + 6.94e4i)10-s − 4.35e5i·11-s + (−2.36e6 − 2.07e4i)12-s + (−4.68e5 + 4.68e5i)13-s + 1.44e6·14-s + (−2.31e6 + 1.80e6i)15-s − 1.57e7·16-s + (2.28e6 − 2.28e6i)17-s + ⋯
L(s)  = 1  + (−1.36 + 1.36i)2-s + (0.00878 − 0.999i)3-s − 2.73i·4-s + (−0.622 − 0.782i)5-s + (1.35 + 1.37i)6-s + (−0.263 − 0.263i)7-s + (2.37 + 2.37i)8-s + (−0.999 − 0.0175i)9-s + (1.92 + 0.219i)10-s − 0.815i·11-s + (−2.73 − 0.0240i)12-s + (−0.350 + 0.350i)13-s + 0.720·14-s + (−0.788 + 0.615i)15-s − 3.76·16-s + (0.390 − 0.390i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 - 0.347i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.937 - 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $-0.937 - 0.347i$
Analytic conductor: \(11.5251\)
Root analytic conductor: \(3.39487\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :11/2),\ -0.937 - 0.347i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.00207972 + 0.0116101i\)
\(L(\frac12)\) \(\approx\) \(0.00207972 + 0.0116101i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-3.69 + 420. i)T \)
5 \( 1 + (4.34e3 + 5.47e3i)T \)
good2 \( 1 + (61.8 - 61.8i)T - 2.04e3iT^{2} \)
7 \( 1 + (1.17e4 + 1.17e4i)T + 1.97e9iT^{2} \)
11 \( 1 + 4.35e5iT - 2.85e11T^{2} \)
13 \( 1 + (4.68e5 - 4.68e5i)T - 1.79e12iT^{2} \)
17 \( 1 + (-2.28e6 + 2.28e6i)T - 3.42e13iT^{2} \)
19 \( 1 - 1.22e6iT - 1.16e14T^{2} \)
23 \( 1 + (-1.59e7 - 1.59e7i)T + 9.52e14iT^{2} \)
29 \( 1 + 1.86e8T + 1.22e16T^{2} \)
31 \( 1 + 3.77e7T + 2.54e16T^{2} \)
37 \( 1 + (-3.83e8 - 3.83e8i)T + 1.77e17iT^{2} \)
41 \( 1 - 5.80e8iT - 5.50e17T^{2} \)
43 \( 1 + (-1.11e8 + 1.11e8i)T - 9.29e17iT^{2} \)
47 \( 1 + (-3.45e8 + 3.45e8i)T - 2.47e18iT^{2} \)
53 \( 1 + (-1.15e9 - 1.15e9i)T + 9.26e18iT^{2} \)
59 \( 1 - 8.18e9T + 3.01e19T^{2} \)
61 \( 1 + 1.00e10T + 4.35e19T^{2} \)
67 \( 1 + (1.32e9 + 1.32e9i)T + 1.22e20iT^{2} \)
71 \( 1 - 3.74e9iT - 2.31e20T^{2} \)
73 \( 1 + (5.37e9 - 5.37e9i)T - 3.13e20iT^{2} \)
79 \( 1 - 1.68e10iT - 7.47e20T^{2} \)
83 \( 1 + (6.15e9 + 6.15e9i)T + 1.28e21iT^{2} \)
89 \( 1 + 1.64e10T + 2.77e21T^{2} \)
97 \( 1 + (5.60e10 + 5.60e10i)T + 7.15e21iT^{2} \)
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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

−16.44037490322955386284820954225, −14.90321966210596578246057898688, −13.44197585100776409581682721461, −11.37772432846442077143508106472, −9.310021158890871253026786708577, −8.132732817662573044501814022355, −7.11883238889606765783020199225, −5.57166247362419288354987207757, −1.17627456546881954424853432786, −0.009462156460010626171788241436, 2.57318716981043552074746954518, 3.88899181271331894971479601230, 7.60766882096313352115847990256, 9.174313386355110662691074971091, 10.29013708215676887225235922833, 11.22983577182087249558027922211, 12.50082389071005065312537387623, 15.00860757396226510801695474902, 16.42647780640794217714480270721, 17.64091092239315856368338457599

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