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Properties

Label 2-15-15.2-c7-0-1
Degree $2$
Conductor $15$
Sign $0.216 - 0.976i$
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  + (6.70 − 6.70i)2-s + (−31.5 + 34.4i)3-s + 38.1i·4-s + (−8.44 + 279. i)5-s + (19.2 + 442. i)6-s + (−68.7 − 68.7i)7-s + (1.11e3 + 1.11e3i)8-s + (−190. − 2.17e3i)9-s + (1.81e3 + 1.92e3i)10-s + 1.96e3i·11-s + (−1.31e3 − 1.20e3i)12-s + (−5.69e3 + 5.69e3i)13-s − 921.·14-s + (−9.36e3 − 9.11e3i)15-s + 1.00e4·16-s + (1.92e4 − 1.92e4i)17-s + ⋯
L(s)  = 1  + (0.592 − 0.592i)2-s + (−0.675 + 0.737i)3-s + 0.298i·4-s + (−0.0302 + 0.999i)5-s + (0.0364 + 0.836i)6-s + (−0.0757 − 0.0757i)7-s + (0.769 + 0.769i)8-s + (−0.0868 − 0.996i)9-s + (0.574 + 0.609i)10-s + 0.444i·11-s + (−0.219 − 0.201i)12-s + (−0.718 + 0.718i)13-s − 0.0897·14-s + (−0.716 − 0.697i)15-s + 0.612·16-s + (0.951 − 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.216 - 0.976i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.216 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.216 - 0.976i$
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: \((2,\ 15,\ (\ :7/2),\ 0.216 - 0.976i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.19765 + 0.960853i\)
\(L(\frac12)\) \(\approx\) \(1.19765 + 0.960853i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (31.5 - 34.4i)T \)
5 \( 1 + (8.44 - 279. i)T \)
good2 \( 1 + (-6.70 + 6.70i)T - 128iT^{2} \)
7 \( 1 + (68.7 + 68.7i)T + 8.23e5iT^{2} \)
11 \( 1 - 1.96e3iT - 1.94e7T^{2} \)
13 \( 1 + (5.69e3 - 5.69e3i)T - 6.27e7iT^{2} \)
17 \( 1 + (-1.92e4 + 1.92e4i)T - 4.10e8iT^{2} \)
19 \( 1 + 2.60e4iT - 8.93e8T^{2} \)
23 \( 1 + (-6.59e4 - 6.59e4i)T + 3.40e9iT^{2} \)
29 \( 1 - 1.39e5T + 1.72e10T^{2} \)
31 \( 1 - 8.27e4T + 2.75e10T^{2} \)
37 \( 1 + (-2.15e5 - 2.15e5i)T + 9.49e10iT^{2} \)
41 \( 1 + 7.07e5iT - 1.94e11T^{2} \)
43 \( 1 + (4.34e5 - 4.34e5i)T - 2.71e11iT^{2} \)
47 \( 1 + (3.80e5 - 3.80e5i)T - 5.06e11iT^{2} \)
53 \( 1 + (-2.37e5 - 2.37e5i)T + 1.17e12iT^{2} \)
59 \( 1 + 4.02e5T + 2.48e12T^{2} \)
61 \( 1 + 1.16e6T + 3.14e12T^{2} \)
67 \( 1 + (4.74e5 + 4.74e5i)T + 6.06e12iT^{2} \)
71 \( 1 + 1.36e6iT - 9.09e12T^{2} \)
73 \( 1 + (-2.19e6 + 2.19e6i)T - 1.10e13iT^{2} \)
79 \( 1 - 4.40e6iT - 1.92e13T^{2} \)
83 \( 1 + (-3.82e6 - 3.82e6i)T + 2.71e13iT^{2} \)
89 \( 1 - 3.52e6T + 4.42e13T^{2} \)
97 \( 1 + (9.16e6 + 9.16e6i)T + 8.07e13iT^{2} \)
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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

−17.84641841758408246228681741739, −16.74552883814387855653488322592, −15.21756784096775593503742020608, −13.88275480404140996998400747370, −12.06758871030746619347474188175, −11.19471586888135831654533236393, −9.764131121753679945765379810311, −7.09710214925356363016052405506, −4.81955073385715000480077877815, −3.08018515229796829530376127672, 0.972112291048169022752771098703, 4.96793476997212152938416995840, 6.16097653388443487617120990312, 7.993491638868394811693335966590, 10.32085940041638522354892699285, 12.29109016970512542094716066046, 13.18385206091675410170040681379, 14.66357654861081515607725361383, 16.26547550205793370274193088520, 17.08455701861080654492968179178

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