↑

Properties

Label 2-15-3.2-c10-0-9
Degree $2$
Conductor $15$
Sign $0.658 + 0.752i$
Analytic cond. $9.53035$
Root an. cond. $3.08712$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4.94i·2-s + (182. − 160. i)3-s + 9.99e2·4-s − 1.39e3i·5-s + (790. + 903. i)6-s − 5.51e3·7-s + 9.99e3i·8-s + (7.79e3 − 5.85e4i)9-s + 6.90e3·10-s − 7.67e4i·11-s + (1.82e5 − 1.60e5i)12-s + 4.49e5·13-s − 2.72e4i·14-s + (−2.23e5 − 2.55e5i)15-s + 9.74e5·16-s − 1.42e6i·17-s + ⋯
L(s)  = 1  + 0.154i·2-s + (0.752 − 0.658i)3-s + 0.976·4-s − 0.447i·5-s + (0.101 + 0.116i)6-s − 0.328·7-s + 0.305i·8-s + (0.131 − 0.991i)9-s + 0.0690·10-s − 0.476i·11-s + (0.734 − 0.643i)12-s + 1.21·13-s − 0.0506i·14-s + (−0.294 − 0.336i)15-s + 0.929·16-s − 1.00i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.658 + 0.752i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.658 + 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.658 + 0.752i$
Analytic conductor: \(9.53035\)
Root analytic conductor: \(3.08712\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :5),\ 0.658 + 0.752i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(2.38328 - 1.08088i\)
\(L(\frac12)\) \(\approx\) \(2.38328 - 1.08088i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-182. + 160. i)T \)
5 \( 1 + 1.39e3iT \)
good2 \( 1 - 4.94iT - 1.02e3T^{2} \)
7 \( 1 + 5.51e3T + 2.82e8T^{2} \)
11 \( 1 + 7.67e4iT - 2.59e10T^{2} \)
13 \( 1 - 4.49e5T + 1.37e11T^{2} \)
17 \( 1 + 1.42e6iT - 2.01e12T^{2} \)
19 \( 1 + 1.48e6T + 6.13e12T^{2} \)
23 \( 1 - 8.24e6iT - 4.14e13T^{2} \)
29 \( 1 - 3.07e7iT - 4.20e14T^{2} \)
31 \( 1 + 1.77e7T + 8.19e14T^{2} \)
37 \( 1 - 7.18e7T + 4.80e15T^{2} \)
41 \( 1 + 2.33e7iT - 1.34e16T^{2} \)
43 \( 1 + 1.37e7T + 2.16e16T^{2} \)
47 \( 1 - 4.03e8iT - 5.25e16T^{2} \)
53 \( 1 - 4.99e8iT - 1.74e17T^{2} \)
59 \( 1 - 7.43e8iT - 5.11e17T^{2} \)
61 \( 1 + 1.54e9T + 7.13e17T^{2} \)
67 \( 1 - 1.83e9T + 1.82e18T^{2} \)
71 \( 1 + 3.45e9iT - 3.25e18T^{2} \)
73 \( 1 - 8.31e8T + 4.29e18T^{2} \)
79 \( 1 - 4.69e8T + 9.46e18T^{2} \)
83 \( 1 - 1.59e9iT - 1.55e19T^{2} \)
89 \( 1 + 6.09e9iT - 3.11e19T^{2} \)
97 \( 1 + 2.89e9T + 7.37e19T^{2} \)
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

−16.48494359168388404266677155594, −15.45180632280677812027511838713, −13.92659019264758989478089030699, −12.65740274927595316353739006501, −11.19536439131664579232937941980, −9.074194445082913289981948390707, −7.60367655109067309015695647376, −6.14639106941958506529895742316, −3.18060726202574314852479056961, −1.35800903504548086155324583343, 2.18533207826646460269541274890, 3.76236182823639145991726241936, 6.40166173092985291935447085512, 8.200871257788894210727330673823, 10.05936043661154204579868611404, 11.10262821289527022655411259395, 12.98593956118860172747645496017, 14.69094190299633982591752835600, 15.57426540133522267464019572660, 16.72599126747585877866573848104

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