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Properties

Label 2-15-15.8-c5-0-3
Degree $2$
Conductor $15$
Sign $0.423 + 0.906i$
Analytic cond. $2.40575$
Root an. cond. $1.55105$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.17 − 3.17i)2-s + (14.5 + 5.54i)3-s − 11.7i·4-s + (11.9 − 54.5i)5-s + (−28.6 − 63.9i)6-s + (83.6 − 83.6i)7-s + (−139. + 139. i)8-s + (181. + 161. i)9-s + (−211. + 135. i)10-s + 566. i·11-s + (65.4 − 171. i)12-s + (−208. − 208. i)13-s − 532.·14-s + (477. − 728. i)15-s + 507.·16-s + (−155. − 155. i)17-s + ⋯
L(s)  = 1  + (−0.561 − 0.561i)2-s + (0.934 + 0.355i)3-s − 0.368i·4-s + (0.214 − 0.976i)5-s + (−0.325 − 0.725i)6-s + (0.645 − 0.645i)7-s + (−0.768 + 0.768i)8-s + (0.746 + 0.665i)9-s + (−0.669 + 0.428i)10-s + 1.41i·11-s + (0.131 − 0.344i)12-s + (−0.342 − 0.342i)13-s − 0.725·14-s + (0.548 − 0.836i)15-s + 0.495·16-s + (−0.130 − 0.130i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.423 + 0.906i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.423 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.423 + 0.906i$
Analytic conductor: \(2.40575\)
Root analytic conductor: \(1.55105\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :5/2),\ 0.423 + 0.906i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.12441 - 0.715778i\)
\(L(\frac12)\) \(\approx\) \(1.12441 - 0.715778i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-14.5 - 5.54i)T \)
5 \( 1 + (-11.9 + 54.5i)T \)
good2 \( 1 + (3.17 + 3.17i)T + 32iT^{2} \)
7 \( 1 + (-83.6 + 83.6i)T - 1.68e4iT^{2} \)
11 \( 1 - 566. iT - 1.61e5T^{2} \)
13 \( 1 + (208. + 208. i)T + 3.71e5iT^{2} \)
17 \( 1 + (155. + 155. i)T + 1.41e6iT^{2} \)
19 \( 1 - 2.30e3iT - 2.47e6T^{2} \)
23 \( 1 + (-1.19e3 + 1.19e3i)T - 6.43e6iT^{2} \)
29 \( 1 - 1.31e3T + 2.05e7T^{2} \)
31 \( 1 - 303.T + 2.86e7T^{2} \)
37 \( 1 + (-2.75e3 + 2.75e3i)T - 6.93e7iT^{2} \)
41 \( 1 + 84.6iT - 1.15e8T^{2} \)
43 \( 1 + (4.68e3 + 4.68e3i)T + 1.47e8iT^{2} \)
47 \( 1 + (1.32e4 + 1.32e4i)T + 2.29e8iT^{2} \)
53 \( 1 + (1.95e4 - 1.95e4i)T - 4.18e8iT^{2} \)
59 \( 1 - 2.08e3T + 7.14e8T^{2} \)
61 \( 1 + 5.08e4T + 8.44e8T^{2} \)
67 \( 1 + (-3.06e4 + 3.06e4i)T - 1.35e9iT^{2} \)
71 \( 1 + 7.52e4iT - 1.80e9T^{2} \)
73 \( 1 + (-1.91e4 - 1.91e4i)T + 2.07e9iT^{2} \)
79 \( 1 - 5.69e4iT - 3.07e9T^{2} \)
83 \( 1 + (-2.55e4 + 2.55e4i)T - 3.93e9iT^{2} \)
89 \( 1 + 4.26e4T + 5.58e9T^{2} \)
97 \( 1 + (-7.80e4 + 7.80e4i)T - 8.58e9iT^{2} \)
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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

−18.24995316186090390507595853633, −16.95446754392209832209729487715, −15.20077693855603302448346031640, −14.11470628267515782945943976605, −12.43696586450231701270952293901, −10.40862615121900900161082313663, −9.433291661001948437164519323712, −7.980665183866949288826191789886, −4.74269706020615530474634882376, −1.71405402104579263455147728297, 2.96100139511524456646099906430, 6.65596068554725203556941533571, 8.085648071610470632615185544381, 9.247631628146367934905028289576, 11.45672812698584607890688557516, 13.34183866999403105152171147809, 14.66051560916519464938811051697, 15.74649519392641988659922652915, 17.51700457938993544996737630254, 18.48212594829250689730638522577

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