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Properties

Label 2-15-15.8-c11-0-2
Degree $2$
Conductor $15$
Sign $0.853 - 0.521i$
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 + (104. − 407. 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 + (2.96e5 + 7.61e4i)12-s + (6.09e5 + 6.09e5i)13-s + 4.31e5·14-s + (2.62e6 + 1.32e6i)15-s + 5.15e6·16-s + (3.57e6 + 3.57e6i)17-s + ⋯
L(s)  = 1  + (−0.823 − 0.823i)2-s + (0.248 − 0.968i)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 + (0.344 + 0.0883i)12-s + (0.455 + 0.455i)13-s + 0.214·14-s + (0.892 + 0.451i)15-s + 1.22·16-s + (0.610 + 0.610i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.853 - 0.521i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.853 - 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.853 - 0.521i$
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: \((2,\ 15,\ (\ :11/2),\ 0.853 - 0.521i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.582500 + 0.164008i\)
\(L(\frac12)\) \(\approx\) \(0.582500 + 0.164008i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-104. + 407. i)T \)
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

−17.53904247603559653300055687422, −15.20755371281542852133661971610, −13.93546863224082541301953003995, −12.17743285369383169488042939996, −11.09938019367136036083798402272, −9.592890081883032764692465052296, −7.974288719223454615321253227252, −6.37935566150571239870016297599, −2.95373348701118924704013596319, −1.54836105560276561446055499275, 0.35112077696970857705331128406, 3.70634689521917267132025911359, 5.69376537084631681380600200005, 8.010230110780031002202449588486, 8.885596672608392190735976256169, 10.23388133396317139460447532106, 12.26645102604664763339192926019, 14.17709157923360474262600150925, 15.92538881871027939847771662193, 16.23254113749453710567795500077

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