↑

Properties

Label 2-15-15.2-c5-0-2
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

Related objects

Downloads

Learn more

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: $\chi_{15} (2, \cdot )$
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 \( 1 + (-7.80e4 - 7.80e4i)T + 8.58e9iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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