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Properties

Label 2-15-15.8-c9-0-1
Degree $2$
Conductor $15$
Sign $0.419 - 0.907i$
Analytic cond. $7.72553$
Root an. cond. $2.77948$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−24.0 − 24.0i)2-s + (139. − 17.1i)3-s + 647. i·4-s + (−1.39e3 − 2.62i)5-s + (−3.76e3 − 2.93e3i)6-s + (−4.15e3 + 4.15e3i)7-s + (3.27e3 − 3.27e3i)8-s + (1.90e4 − 4.78e3i)9-s + (3.35e4 + 3.37e4i)10-s + 5.33e4i·11-s + (1.11e4 + 9.02e4i)12-s + (2.32e4 + 2.32e4i)13-s + 2.00e5·14-s + (−1.94e5 + 2.36e4i)15-s + 1.74e5·16-s + (−2.52e5 − 2.52e5i)17-s + ⋯
L(s)  = 1  + (−1.06 − 1.06i)2-s + (0.992 − 0.122i)3-s + 1.26i·4-s + (−0.999 − 0.00187i)5-s + (−1.18 − 0.925i)6-s + (−0.653 + 0.653i)7-s + (0.282 − 0.282i)8-s + (0.969 − 0.243i)9-s + (1.06 + 1.06i)10-s + 1.09i·11-s + (0.154 + 1.25i)12-s + (0.226 + 0.226i)13-s + 1.39·14-s + (−0.992 + 0.120i)15-s + 0.664·16-s + (−0.732 − 0.732i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.419 - 0.907i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.419 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.419 - 0.907i$
Analytic conductor: \(7.72553\)
Root analytic conductor: \(2.77948\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :9/2),\ 0.419 - 0.907i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.411948 + 0.263509i\)
\(L(\frac12)\) \(\approx\) \(0.411948 + 0.263509i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-139. + 17.1i)T \)
5 \( 1 + (1.39e3 + 2.62i)T \)
good2 \( 1 + (24.0 + 24.0i)T + 512iT^{2} \)
7 \( 1 + (4.15e3 - 4.15e3i)T - 4.03e7iT^{2} \)
11 \( 1 - 5.33e4iT - 2.35e9T^{2} \)
13 \( 1 + (-2.32e4 - 2.32e4i)T + 1.06e10iT^{2} \)
17 \( 1 + (2.52e5 + 2.52e5i)T + 1.18e11iT^{2} \)
19 \( 1 - 5.88e5iT - 3.22e11T^{2} \)
23 \( 1 + (1.34e6 - 1.34e6i)T - 1.80e12iT^{2} \)
29 \( 1 + 1.67e6T + 1.45e13T^{2} \)
31 \( 1 + 9.03e6T + 2.64e13T^{2} \)
37 \( 1 + (-1.99e6 + 1.99e6i)T - 1.29e14iT^{2} \)
41 \( 1 - 1.85e7iT - 3.27e14T^{2} \)
43 \( 1 + (2.46e7 + 2.46e7i)T + 5.02e14iT^{2} \)
47 \( 1 + (2.25e6 + 2.25e6i)T + 1.11e15iT^{2} \)
53 \( 1 + (-6.59e6 + 6.59e6i)T - 3.29e15iT^{2} \)
59 \( 1 - 2.19e7T + 8.66e15T^{2} \)
61 \( 1 - 9.47e7T + 1.16e16T^{2} \)
67 \( 1 + (2.46e6 - 2.46e6i)T - 2.72e16iT^{2} \)
71 \( 1 - 3.48e8iT - 4.58e16T^{2} \)
73 \( 1 + (1.28e8 + 1.28e8i)T + 5.88e16iT^{2} \)
79 \( 1 + 2.62e8iT - 1.19e17T^{2} \)
83 \( 1 + (-1.32e8 + 1.32e8i)T - 1.86e17iT^{2} \)
89 \( 1 - 9.05e8T + 3.50e17T^{2} \)
97 \( 1 + (-4.92e8 + 4.92e8i)T - 7.60e17iT^{2} \)
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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

−18.14399016198052618063061431180, −16.07474166184600712354332451426, −14.85375806056265133277185350736, −12.77025053029239354511963269991, −11.70495703800155750652580199356, −9.899322111826311383772996416149, −8.872946652246016218489546046278, −7.50471030529221958114172649337, −3.59347170863873416345649597689, −1.97857511064108769556159995222, 0.31429483618337646058762874408, 3.65329348760096714848777419043, 6.75735509775559921937313119331, 8.034052833255121152095644035610, 8.991447568697417775988328608616, 10.63746575658259747145189430705, 13.10158380053492355388995848797, 14.73440973590791930080270551190, 15.85290132075802326429564187346, 16.55410866909293785279376155161

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