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Properties

Label 2-15-15.8-c11-0-12
Degree $2$
Conductor $15$
Sign $0.867 + 0.496i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (37.2 + 37.2i)2-s + (−407. + 104. i)3-s + 728. i·4-s + (1.50e3 − 6.82e3i)5-s + (−1.90e4 − 1.12e4i)6-s + (−5.79e3 + 5.79e3i)7-s + (4.91e4 − 4.91e4i)8-s + (1.55e5 − 8.52e4i)9-s + (3.10e5 − 1.98e5i)10-s − 3.64e5i·11-s + (−7.61e4 − 2.96e5i)12-s + (6.09e5 + 6.09e5i)13-s − 4.31e5·14-s + (9.88e4 + 2.93e6i)15-s + 5.15e6·16-s + (−3.57e6 − 3.57e6i)17-s + ⋯
L(s)  = 1  + (0.823 + 0.823i)2-s + (−0.968 + 0.248i)3-s + 0.355i·4-s + (0.215 − 0.976i)5-s + (−1.00 − 0.593i)6-s + (−0.130 + 0.130i)7-s + (0.530 − 0.530i)8-s + (0.876 − 0.481i)9-s + (0.981 − 0.626i)10-s − 0.682i·11-s + (−0.0883 − 0.344i)12-s + (0.455 + 0.455i)13-s − 0.214·14-s + (0.0335 + 0.999i)15-s + 1.22·16-s + (−0.610 − 0.610i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.867 + 0.496i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.867 + 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.867 + 0.496i$
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: \((2,\ 15,\ (\ :11/2),\ 0.867 + 0.496i)\)

Particular Values

\(L(6)\) \(\approx\) \(1.84588 - 0.491017i\)
\(L(\frac12)\) \(\approx\) \(1.84588 - 0.491017i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (407. - 104. i)T \)
5 \( 1 + (-1.50e3 + 6.82e3i)T \)
good2 \( 1 + (-37.2 - 37.2i)T + 2.04e3iT^{2} \)
7 \( 1 + (5.79e3 - 5.79e3i)T - 1.97e9iT^{2} \)
11 \( 1 + 3.64e5iT - 2.85e11T^{2} \)
13 \( 1 + (-6.09e5 - 6.09e5i)T + 1.79e12iT^{2} \)
17 \( 1 + (3.57e6 + 3.57e6i)T + 3.42e13iT^{2} \)
19 \( 1 + 1.30e7iT - 1.16e14T^{2} \)
23 \( 1 + (-3.07e7 + 3.07e7i)T - 9.52e14iT^{2} \)
29 \( 1 + 9.62e7T + 1.22e16T^{2} \)
31 \( 1 + 1.54e7T + 2.54e16T^{2} \)
37 \( 1 + (4.15e8 - 4.15e8i)T - 1.77e17iT^{2} \)
41 \( 1 + 1.11e9iT - 5.50e17T^{2} \)
43 \( 1 + (-1.03e9 - 1.03e9i)T + 9.29e17iT^{2} \)
47 \( 1 + (-1.26e9 - 1.26e9i)T + 2.47e18iT^{2} \)
53 \( 1 + (-1.85e9 + 1.85e9i)T - 9.26e18iT^{2} \)
59 \( 1 - 3.23e9T + 3.01e19T^{2} \)
61 \( 1 + 5.92e9T + 4.35e19T^{2} \)
67 \( 1 + (-8.83e9 + 8.83e9i)T - 1.22e20iT^{2} \)
71 \( 1 - 1.82e10iT - 2.31e20T^{2} \)
73 \( 1 + (-3.15e9 - 3.15e9i)T + 3.13e20iT^{2} \)
79 \( 1 - 2.48e10iT - 7.47e20T^{2} \)
83 \( 1 + (-2.65e10 + 2.65e10i)T - 1.28e21iT^{2} \)
89 \( 1 + 5.83e10T + 2.77e21T^{2} \)
97 \( 1 + (9.17e10 - 9.17e10i)T - 7.15e21iT^{2} \)
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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

−16.28393994698139078216159191311, −15.51057023145411855923884010862, −13.71168011746434934653667359197, −12.66496361946185641947214698689, −11.02651367546557180252009044875, −9.150020732061078986261111792488, −6.77450273538087808435612380673, −5.49586152756892894259963206596, −4.41342013132201635818604458207, −0.78073728915845354542076386305, 1.85184195875273197034248123205, 3.77536180716284062532214081818, 5.63883262527588495053552098021, 7.32702945772187002006916783845, 10.32510338313396074608716779280, 11.21718854087629456839353544429, 12.50627856038936064733750360800, 13.57202806437101263904972801355, 15.15272782868472497417740643873, 16.97533697634990676529491331192

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