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Properties

Label 2-15-15.14-c4-0-2
Degree $2$
Conductor $15$
Sign $0.955 - 0.294i$
Analytic cond. $1.55054$
Root an. cond. $1.24521$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.16·2-s + (4.74 + 7.64i)3-s − 5.99·4-s + (6.32 − 24.1i)5-s + (15.0 + 24.1i)6-s − 15.2i·7-s − 69.5·8-s + (−36.0 + 72.5i)9-s + (20.0 − 76.4i)10-s − 96.7i·11-s + (−28.4 − 45.8i)12-s + 244. i·13-s − 48.3i·14-s + (214. − 66.3i)15-s − 124.·16-s + 278.·17-s + ⋯
L(s)  = 1  + 0.790·2-s + (0.527 + 0.849i)3-s − 0.374·4-s + (0.252 − 0.967i)5-s + (0.416 + 0.671i)6-s − 0.312i·7-s − 1.08·8-s + (−0.444 + 0.895i)9-s + (0.200 − 0.764i)10-s − 0.799i·11-s + (−0.197 − 0.318i)12-s + 1.44i·13-s − 0.246i·14-s + (0.955 − 0.294i)15-s − 0.484·16-s + 0.962·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.294i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.955 - 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.955 - 0.294i$
Analytic conductor: \(1.55054\)
Root analytic conductor: \(1.24521\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :2),\ 0.955 - 0.294i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.63504 + 0.246577i\)
\(L(\frac12)\) \(\approx\) \(1.63504 + 0.246577i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.74 - 7.64i)T \)
5 \( 1 + (-6.32 + 24.1i)T \)
good2 \( 1 - 3.16T + 16T^{2} \)
7 \( 1 + 15.2iT - 2.40e3T^{2} \)
11 \( 1 + 96.7iT - 1.46e4T^{2} \)
13 \( 1 - 244. iT - 2.85e4T^{2} \)
17 \( 1 - 278.T + 8.35e4T^{2} \)
19 \( 1 - 308T + 1.30e5T^{2} \)
23 \( 1 + 414.T + 2.79e5T^{2} \)
29 \( 1 - 193. iT - 7.07e5T^{2} \)
31 \( 1 - 32T + 9.23e5T^{2} \)
37 \( 1 - 1.28e3iT - 1.87e6T^{2} \)
41 \( 1 + 2.08e3iT - 2.82e6T^{2} \)
43 \( 1 + 2.58e3iT - 3.41e6T^{2} \)
47 \( 1 + 2.44e3T + 4.87e6T^{2} \)
53 \( 1 - 1.41e3T + 7.89e6T^{2} \)
59 \( 1 + 3.96e3iT - 1.21e7T^{2} \)
61 \( 1 + 928T + 1.38e7T^{2} \)
67 \( 1 - 2.58e3iT - 2.01e7T^{2} \)
71 \( 1 - 4.64e3iT - 2.54e7T^{2} \)
73 \( 1 - 4.22e3iT - 2.83e7T^{2} \)
79 \( 1 - 8T + 3.89e7T^{2} \)
83 \( 1 + 4.43e3T + 4.74e7T^{2} \)
89 \( 1 - 9.28e3iT - 6.27e7T^{2} \)
97 \( 1 - 2.50e3iT - 8.85e7T^{2} \)
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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

−18.87618686040980262372545696617, −16.95081012828054957580361749229, −15.95143369593200129511075512075, −14.21576245897445210362723740511, −13.63206130947250369010704726116, −11.90644124478148003091507148755, −9.721460416958051393786941291175, −8.579577084164756627338195895756, −5.37193848658419878445592533012, −3.90366100692623750862261309137, 3.07872863286147867877675312894, 5.87413455040335138768004992486, 7.75234641351409789087364819982, 9.766299278577160799377346417051, 12.05348935887108114357165880236, 13.18488156434099070577097185574, 14.37345708194104436932401551287, 15.16006712389052846824721981193, 17.88377707753321760602638546250, 18.27628515125553774618298378779

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