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Properties

Label 2-15-15.8-c11-0-8
Degree $2$
Conductor $15$
Sign $-0.974 + 0.222i$
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  + (57.5 + 57.5i)2-s + (−239. + 345. i)3-s + 4.57e3i·4-s + (−1.29e3 + 6.86e3i)5-s + (−3.37e4 + 6.10e3i)6-s + (5.37e4 − 5.37e4i)7-s + (−1.45e5 + 1.45e5i)8-s + (−6.21e4 − 1.65e5i)9-s + (−4.69e5 + 3.20e5i)10-s + 2.99e5i·11-s + (−1.58e6 − 1.09e6i)12-s + (2.57e5 + 2.57e5i)13-s + 6.19e6·14-s + (−2.06e6 − 2.09e6i)15-s − 7.38e6·16-s + (−1.91e6 − 1.91e6i)17-s + ⋯
L(s)  = 1  + (1.27 + 1.27i)2-s + (−0.569 + 0.821i)3-s + 2.23i·4-s + (−0.185 + 0.982i)5-s + (−1.76 + 0.320i)6-s + (1.20 − 1.20i)7-s + (−1.57 + 1.57i)8-s + (−0.350 − 0.936i)9-s + (−1.48 + 1.01i)10-s + 0.559i·11-s + (−1.83 − 1.27i)12-s + (0.192 + 0.192i)13-s + 3.07·14-s + (−0.701 − 0.712i)15-s − 1.76·16-s + (−0.326 − 0.326i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.222i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.974 + 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $-0.974 + 0.222i$
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: \((2,\ 15,\ (\ :11/2),\ -0.974 + 0.222i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.317893 - 2.81918i\)
\(L(\frac12)\) \(\approx\) \(0.317893 - 2.81918i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (239. - 345. i)T \)
5 \( 1 + (1.29e3 - 6.86e3i)T \)
good2 \( 1 + (-57.5 - 57.5i)T + 2.04e3iT^{2} \)
7 \( 1 + (-5.37e4 + 5.37e4i)T - 1.97e9iT^{2} \)
11 \( 1 - 2.99e5iT - 2.85e11T^{2} \)
13 \( 1 + (-2.57e5 - 2.57e5i)T + 1.79e12iT^{2} \)
17 \( 1 + (1.91e6 + 1.91e6i)T + 3.42e13iT^{2} \)
19 \( 1 - 1.22e7iT - 1.16e14T^{2} \)
23 \( 1 + (-1.69e4 + 1.69e4i)T - 9.52e14iT^{2} \)
29 \( 1 - 8.47e7T + 1.22e16T^{2} \)
31 \( 1 - 5.68e7T + 2.54e16T^{2} \)
37 \( 1 + (2.28e8 - 2.28e8i)T - 1.77e17iT^{2} \)
41 \( 1 + 4.14e8iT - 5.50e17T^{2} \)
43 \( 1 + (-5.52e8 - 5.52e8i)T + 9.29e17iT^{2} \)
47 \( 1 + (-3.42e8 - 3.42e8i)T + 2.47e18iT^{2} \)
53 \( 1 + (-1.12e9 + 1.12e9i)T - 9.26e18iT^{2} \)
59 \( 1 + 6.76e9T + 3.01e19T^{2} \)
61 \( 1 - 3.58e9T + 4.35e19T^{2} \)
67 \( 1 + (-1.15e10 + 1.15e10i)T - 1.22e20iT^{2} \)
71 \( 1 + 2.04e10iT - 2.31e20T^{2} \)
73 \( 1 + (-8.20e9 - 8.20e9i)T + 3.13e20iT^{2} \)
79 \( 1 + 2.87e10iT - 7.47e20T^{2} \)
83 \( 1 + (6.21e9 - 6.21e9i)T - 1.28e21iT^{2} \)
89 \( 1 - 9.02e10T + 2.77e21T^{2} \)
97 \( 1 + (1.89e9 - 1.89e9i)T - 7.15e21iT^{2} \)
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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

−16.98315966219174776257640061096, −15.72009773349105926065858152816, −14.64949970342150513415749544711, −13.95371417413842019967575243129, −11.88903655708168445620041873480, −10.55086538310197777788088249064, −7.77120211002359865250709679827, −6.50542891907607336883770921800, −4.80973544612508459086778617204, −3.77146520184955631693831208561, 0.987374217381183612042278350246, 2.28323311357639221707273704640, 4.76781796490301810744079635675, 5.71542398671712336330828872348, 8.570390015508218684018332116212, 11.08086921460681726270684930132, 11.87637084419655589962214247517, 12.78993860020197361220905337250, 13.93886127631810039385746882715, 15.51872938953097640862377318785

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