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Properties

Label 2-15-15.8-c11-0-7
Degree $2$
Conductor $15$
Sign $0.872 - 0.487i$
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  + (−37.4 − 37.4i)2-s + (369. + 201. i)3-s + 763. i·4-s + (6.98e3 + 67.8i)5-s + (−6.32e3 − 2.13e4i)6-s + (−4.63e4 + 4.63e4i)7-s + (−4.81e4 + 4.81e4i)8-s + (9.63e4 + 1.48e5i)9-s + (−2.59e5 − 2.64e5i)10-s − 5.56e4i·11-s + (−1.53e5 + 2.82e5i)12-s + (6.31e5 + 6.31e5i)13-s + 3.47e6·14-s + (2.57e6 + 1.42e6i)15-s + 5.17e6·16-s + (2.79e6 + 2.79e6i)17-s + ⋯
L(s)  = 1  + (−0.828 − 0.828i)2-s + (0.878 + 0.477i)3-s + 0.372i·4-s + (0.999 + 0.00970i)5-s + (−0.332 − 1.12i)6-s + (−1.04 + 1.04i)7-s + (−0.519 + 0.519i)8-s + (0.543 + 0.839i)9-s + (−0.820 − 0.836i)10-s − 0.104i·11-s + (−0.177 + 0.327i)12-s + (0.471 + 0.471i)13-s + 1.72·14-s + (0.873 + 0.486i)15-s + 1.23·16-s + (0.477 + 0.477i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 - 0.487i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.872 - 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.872 - 0.487i$
Analytic conductor: \(11.5251\)
Root analytic conductor: \(3.39487\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :11/2),\ 0.872 - 0.487i)\)

Particular Values

\(L(6)\) \(\approx\) \(1.45460 + 0.378888i\)
\(L(\frac12)\) \(\approx\) \(1.45460 + 0.378888i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-369. - 201. i)T \)
5 \( 1 + (-6.98e3 - 67.8i)T \)
good2 \( 1 + (37.4 + 37.4i)T + 2.04e3iT^{2} \)
7 \( 1 + (4.63e4 - 4.63e4i)T - 1.97e9iT^{2} \)
11 \( 1 + 5.56e4iT - 2.85e11T^{2} \)
13 \( 1 + (-6.31e5 - 6.31e5i)T + 1.79e12iT^{2} \)
17 \( 1 + (-2.79e6 - 2.79e6i)T + 3.42e13iT^{2} \)
19 \( 1 - 1.01e7iT - 1.16e14T^{2} \)
23 \( 1 + (-1.26e7 + 1.26e7i)T - 9.52e14iT^{2} \)
29 \( 1 + 1.48e8T + 1.22e16T^{2} \)
31 \( 1 - 2.61e8T + 2.54e16T^{2} \)
37 \( 1 + (3.41e8 - 3.41e8i)T - 1.77e17iT^{2} \)
41 \( 1 + 1.32e9iT - 5.50e17T^{2} \)
43 \( 1 + (6.29e7 + 6.29e7i)T + 9.29e17iT^{2} \)
47 \( 1 + (-1.38e9 - 1.38e9i)T + 2.47e18iT^{2} \)
53 \( 1 + (4.62e8 - 4.62e8i)T - 9.26e18iT^{2} \)
59 \( 1 + 6.47e9T + 3.01e19T^{2} \)
61 \( 1 + 6.68e9T + 4.35e19T^{2} \)
67 \( 1 + (-3.32e9 + 3.32e9i)T - 1.22e20iT^{2} \)
71 \( 1 - 7.73e9iT - 2.31e20T^{2} \)
73 \( 1 + (-1.89e9 - 1.89e9i)T + 3.13e20iT^{2} \)
79 \( 1 + 2.39e10iT - 7.47e20T^{2} \)
83 \( 1 + (2.08e10 - 2.08e10i)T - 1.28e21iT^{2} \)
89 \( 1 - 1.75e10T + 2.77e21T^{2} \)
97 \( 1 + (-4.63e10 + 4.63e10i)T - 7.15e21iT^{2} \)
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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

−16.91735946699827132017444636088, −15.37937342700163621858388193653, −13.99879734187124207621070559778, −12.45836406283078529453093422529, −10.46919024667229679053622845585, −9.509569726720401332167976404191, −8.677845213792552335538505154679, −5.88547186992241312184152775044, −3.04869363229047561165430437326, −1.79012575886273183779565206153, 0.825730642016448694184497938419, 3.18428211617689648773748002246, 6.41588465974061919588938239326, 7.46429855053449438113930481584, 9.085796820173548011376372274665, 10.01219603307512800320794143310, 12.90268268240054987149655765138, 13.74248611289257763075141694470, 15.38240152908191487278781062396, 16.72240819619630554551205988487

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