Properties

Label 2-15-15.8-c7-0-4
Degree $2$
Conductor $15$
Sign $0.861 + 0.508i$
Analytic cond. $4.68577$
Root an. cond. $2.16466$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−13.2 − 13.2i)2-s + (42.4 + 19.6i)3-s + 224. i·4-s + (122. + 251. i)5-s + (−301. − 824. i)6-s + (796. − 796. i)7-s + (1.27e3 − 1.27e3i)8-s + (1.41e3 + 1.67e3i)9-s + (1.70e3 − 4.96e3i)10-s − 2.19e3i·11-s + (−4.41e3 + 9.50e3i)12-s + (5.05e3 + 5.05e3i)13-s − 2.11e4·14-s + (268. + 1.30e4i)15-s − 5.12e3·16-s + (1.73e4 + 1.73e4i)17-s + ⋯
L(s)  = 1  + (−1.17 − 1.17i)2-s + (0.907 + 0.421i)3-s + 1.75i·4-s + (0.439 + 0.898i)5-s + (−0.569 − 1.55i)6-s + (0.877 − 0.877i)7-s + (0.879 − 0.879i)8-s + (0.645 + 0.763i)9-s + (0.537 − 1.56i)10-s − 0.497i·11-s + (−0.737 + 1.58i)12-s + (0.637 + 0.637i)13-s − 2.05·14-s + (0.0205 + 0.999i)15-s − 0.313·16-s + (0.856 + 0.856i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 + 0.508i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.861 + 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.861 + 0.508i$
Analytic conductor: \(4.68577\)
Root analytic conductor: \(2.16466\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :7/2),\ 0.861 + 0.508i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.24456 - 0.339785i\)
\(L(\frac12)\) \(\approx\) \(1.24456 - 0.339785i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-42.4 - 19.6i)T \)
5 \( 1 + (-122. - 251. i)T \)
good2 \( 1 + (13.2 + 13.2i)T + 128iT^{2} \)
7 \( 1 + (-796. + 796. i)T - 8.23e5iT^{2} \)
11 \( 1 + 2.19e3iT - 1.94e7T^{2} \)
13 \( 1 + (-5.05e3 - 5.05e3i)T + 6.27e7iT^{2} \)
17 \( 1 + (-1.73e4 - 1.73e4i)T + 4.10e8iT^{2} \)
19 \( 1 + 3.65e4iT - 8.93e8T^{2} \)
23 \( 1 + (9.55e3 - 9.55e3i)T - 3.40e9iT^{2} \)
29 \( 1 + 3.53e4T + 1.72e10T^{2} \)
31 \( 1 + 2.37e5T + 2.75e10T^{2} \)
37 \( 1 + (-6.41e4 + 6.41e4i)T - 9.49e10iT^{2} \)
41 \( 1 - 9.92e3iT - 1.94e11T^{2} \)
43 \( 1 + (1.90e5 + 1.90e5i)T + 2.71e11iT^{2} \)
47 \( 1 + (2.13e5 + 2.13e5i)T + 5.06e11iT^{2} \)
53 \( 1 + (-8.59e5 + 8.59e5i)T - 1.17e12iT^{2} \)
59 \( 1 + 1.97e6T + 2.48e12T^{2} \)
61 \( 1 + 1.08e6T + 3.14e12T^{2} \)
67 \( 1 + (3.35e5 - 3.35e5i)T - 6.06e12iT^{2} \)
71 \( 1 + 4.54e6iT - 9.09e12T^{2} \)
73 \( 1 + (5.22e5 + 5.22e5i)T + 1.10e13iT^{2} \)
79 \( 1 + 3.93e6iT - 1.92e13T^{2} \)
83 \( 1 + (-4.46e6 + 4.46e6i)T - 2.71e13iT^{2} \)
89 \( 1 + 7.73e6T + 4.42e13T^{2} \)
97 \( 1 + (-8.72e4 + 8.72e4i)T - 8.07e13iT^{2} \)
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

−18.04723111308664867270502609693, −16.75521415849831196341805908589, −14.70788751107575073420285380036, −13.51192700019890446298276312324, −11.17218209688777230541535450376, −10.43711039501609020794994492502, −9.056199825005170818425840216035, −7.64204763123215546821764165443, −3.52719848652555004086928971180, −1.74201890473815851791497956155, 1.44036778915476516941523407589, 5.65778465839247316875433923168, 7.73215980750577443755854431714, 8.629817626464556216259730349149, 9.725045371180229889528683278323, 12.45597270381401719997765159717, 14.25706014731598430447784776093, 15.31746624434243451814345243820, 16.58491372824102155411740715915, 18.05417702343585075685750773943

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