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Properties

Label 2-15-15.2-c7-0-10
Degree $2$
Conductor $15$
Sign $-0.882 + 0.470i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (9.36 − 9.36i)2-s + (−40.3 − 23.6i)3-s − 47.4i·4-s + (−125. − 249. i)5-s + (−599. + 155. i)6-s + (−696. − 696. i)7-s + (754. + 754. i)8-s + (1.06e3 + 1.91e3i)9-s + (−3.51e3 − 1.15e3i)10-s − 3.59e3i·11-s + (−1.12e3 + 1.91e3i)12-s + (7.09e3 − 7.09e3i)13-s − 1.30e4·14-s + (−832. + 1.30e4i)15-s + 2.02e4·16-s + (1.22e4 − 1.22e4i)17-s + ⋯
L(s)  = 1  + (0.827 − 0.827i)2-s + (−0.862 − 0.506i)3-s − 0.371i·4-s + (−0.450 − 0.892i)5-s + (−1.13 + 0.294i)6-s + (−0.767 − 0.767i)7-s + (0.520 + 0.520i)8-s + (0.486 + 0.873i)9-s + (−1.11 − 0.366i)10-s − 0.813i·11-s + (−0.187 + 0.319i)12-s + (0.895 − 0.895i)13-s − 1.27·14-s + (−0.0637 + 0.997i)15-s + 1.23·16-s + (0.606 − 0.606i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.882 + 0.470i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.882 + 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $-0.882 + 0.470i$
Analytic conductor: \(4.68577\)
Root analytic conductor: \(2.16466\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :7/2),\ -0.882 + 0.470i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.349604 - 1.39882i\)
\(L(\frac12)\) \(\approx\) \(0.349604 - 1.39882i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (40.3 + 23.6i)T \)
5 \( 1 + (125. + 249. i)T \)
good2 \( 1 + (-9.36 + 9.36i)T - 128iT^{2} \)
7 \( 1 + (696. + 696. i)T + 8.23e5iT^{2} \)
11 \( 1 + 3.59e3iT - 1.94e7T^{2} \)
13 \( 1 + (-7.09e3 + 7.09e3i)T - 6.27e7iT^{2} \)
17 \( 1 + (-1.22e4 + 1.22e4i)T - 4.10e8iT^{2} \)
19 \( 1 - 2.99e4iT - 8.93e8T^{2} \)
23 \( 1 + (2.11e4 + 2.11e4i)T + 3.40e9iT^{2} \)
29 \( 1 + 1.26e5T + 1.72e10T^{2} \)
31 \( 1 + 4.59e3T + 2.75e10T^{2} \)
37 \( 1 + (-2.98e5 - 2.98e5i)T + 9.49e10iT^{2} \)
41 \( 1 + 4.53e5iT - 1.94e11T^{2} \)
43 \( 1 + (-2.39e5 + 2.39e5i)T - 2.71e11iT^{2} \)
47 \( 1 + (1.01e5 - 1.01e5i)T - 5.06e11iT^{2} \)
53 \( 1 + (-1.14e6 - 1.14e6i)T + 1.17e12iT^{2} \)
59 \( 1 - 5.24e5T + 2.48e12T^{2} \)
61 \( 1 - 2.65e6T + 3.14e12T^{2} \)
67 \( 1 + (2.03e6 + 2.03e6i)T + 6.06e12iT^{2} \)
71 \( 1 - 1.13e6iT - 9.09e12T^{2} \)
73 \( 1 + (-3.59e6 + 3.59e6i)T - 1.10e13iT^{2} \)
79 \( 1 - 1.21e6iT - 1.92e13T^{2} \)
83 \( 1 + (3.30e6 + 3.30e6i)T + 2.71e13iT^{2} \)
89 \( 1 + 4.24e6T + 4.42e13T^{2} \)
97 \( 1 + (-8.69e6 - 8.69e6i)T + 8.07e13iT^{2} \)
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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

−16.90791344874216866810946390719, −16.20661570854583750934438701957, −13.63328404579657102165277253490, −12.86410781053175817608675771777, −11.79867321055922686788319681657, −10.52415430194156995731619429198, −7.88806118137945318026789066898, −5.61222333276837893524243332011, −3.77002835352317476367317098620, −0.850321290753286854393050041719, 3.99312673984192232266893350448, 5.86721270110615463896935856932, 6.93216735394640660800505502360, 9.773405848854951751175204063952, 11.35605548721368883038569952502, 12.86387753763423600541153860044, 14.68579776988349895544607841580, 15.54401028476344841710881599038, 16.39938931646710527028513211444, 18.15102749686615286538639810627

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