↑

Properties

Label 2-15-15.8-c11-0-4
Degree $2$
Conductor $15$
Sign $0.344 - 0.938i$
Analytic cond. $11.5251$
Root an. cond. $3.39487$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (3.46 + 3.46i)2-s + (−402. − 124. i)3-s − 2.02e3i·4-s + (−6.12e3 + 3.36e3i)5-s + (−961. − 1.82e3i)6-s + (1.06e4 − 1.06e4i)7-s + (1.40e4 − 1.40e4i)8-s + (1.46e5 + 1.00e5i)9-s + (−3.28e4 − 9.56e3i)10-s + 4.67e5i·11-s + (−2.51e5 + 8.13e5i)12-s + (2.91e5 + 2.91e5i)13-s + 7.36e4·14-s + (2.88e6 − 5.89e5i)15-s − 4.04e6·16-s + (4.15e6 + 4.15e6i)17-s + ⋯
L(s)  = 1  + (0.0764 + 0.0764i)2-s + (−0.955 − 0.295i)3-s − 0.988i·4-s + (−0.876 + 0.481i)5-s + (−0.0504 − 0.0956i)6-s + (0.239 − 0.239i)7-s + (0.152 − 0.152i)8-s + (0.825 + 0.564i)9-s + (−0.103 − 0.0302i)10-s + 0.874i·11-s + (−0.292 + 0.944i)12-s + (0.218 + 0.218i)13-s + 0.0365·14-s + (0.979 − 0.200i)15-s − 0.965·16-s + (0.710 + 0.710i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.344 - 0.938i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.344 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.344 - 0.938i$
Analytic conductor: \(11.5251\)
Root analytic conductor: \(3.39487\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :11/2),\ 0.344 - 0.938i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.636698 + 0.444570i\)
\(L(\frac12)\) \(\approx\) \(0.636698 + 0.444570i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (402. + 124. i)T \)
5 \( 1 + (6.12e3 - 3.36e3i)T \)
good2 \( 1 + (-3.46 - 3.46i)T + 2.04e3iT^{2} \)
7 \( 1 + (-1.06e4 + 1.06e4i)T - 1.97e9iT^{2} \)
11 \( 1 - 4.67e5iT - 2.85e11T^{2} \)
13 \( 1 + (-2.91e5 - 2.91e5i)T + 1.79e12iT^{2} \)
17 \( 1 + (-4.15e6 - 4.15e6i)T + 3.42e13iT^{2} \)
19 \( 1 - 1.34e7iT - 1.16e14T^{2} \)
23 \( 1 + (-1.58e7 + 1.58e7i)T - 9.52e14iT^{2} \)
29 \( 1 + 1.33e8T + 1.22e16T^{2} \)
31 \( 1 - 9.01e7T + 2.54e16T^{2} \)
37 \( 1 + (7.49e7 - 7.49e7i)T - 1.77e17iT^{2} \)
41 \( 1 - 1.18e9iT - 5.50e17T^{2} \)
43 \( 1 + (4.06e8 + 4.06e8i)T + 9.29e17iT^{2} \)
47 \( 1 + (-1.92e9 - 1.92e9i)T + 2.47e18iT^{2} \)
53 \( 1 + (2.90e9 - 2.90e9i)T - 9.26e18iT^{2} \)
59 \( 1 + 5.75e9T + 3.01e19T^{2} \)
61 \( 1 - 4.98e9T + 4.35e19T^{2} \)
67 \( 1 + (-3.09e9 + 3.09e9i)T - 1.22e20iT^{2} \)
71 \( 1 - 1.59e10iT - 2.31e20T^{2} \)
73 \( 1 + (2.43e10 + 2.43e10i)T + 3.13e20iT^{2} \)
79 \( 1 - 4.56e10iT - 7.47e20T^{2} \)
83 \( 1 + (-1.59e10 + 1.59e10i)T - 1.28e21iT^{2} \)
89 \( 1 + 4.94e10T + 2.77e21T^{2} \)
97 \( 1 + (-1.06e11 + 1.06e11i)T - 7.15e21iT^{2} \)
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

−16.86643261364344184057132902196, −15.49371460835654937430388137351, −14.39970078413159302458593603160, −12.50761354888556479680546822229, −11.16387825053666543267093270642, −10.13473814452666722007504581606, −7.54586373503118843932418301673, −6.13492737268861159499144575673, −4.44880971610806294377005840934, −1.35786478989607185641449818179, 0.42789487689413214075983048724, 3.55985713703983514113665227693, 5.12992631678958378528216374836, 7.32177798103689084867721017863, 8.880863815472112757716093821048, 11.18026482852377001433237204285, 11.93016238266142639346899899514, 13.21250378795646072276932409217, 15.51364774788533120622626358579, 16.42144663736211663790640507381

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