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Properties

Label 2-15-15.8-c9-0-9
Degree $2$
Conductor $15$
Sign $-0.866 + 0.498i$
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  + (−24.1 − 24.1i)2-s + (21.4 + 138. i)3-s + 652. i·4-s + (844. − 1.11e3i)5-s + (2.82e3 − 3.86e3i)6-s + (1.78e3 − 1.78e3i)7-s + (3.38e3 − 3.38e3i)8-s + (−1.87e4 + 5.95e3i)9-s + (−4.72e4 + 6.47e3i)10-s − 2.05e4i·11-s + (−9.04e4 + 1.40e4i)12-s + (−1.05e5 − 1.05e5i)13-s − 8.59e4·14-s + (1.72e5 + 9.32e4i)15-s + 1.70e5·16-s + (−2.69e5 − 2.69e5i)17-s + ⋯
L(s)  = 1  + (−1.06 − 1.06i)2-s + (0.153 + 0.988i)3-s + 1.27i·4-s + (0.604 − 0.796i)5-s + (0.890 − 1.21i)6-s + (0.280 − 0.280i)7-s + (0.291 − 0.291i)8-s + (−0.953 + 0.302i)9-s + (−1.49 + 0.204i)10-s − 0.423i·11-s + (−1.25 + 0.195i)12-s + (−1.02 − 1.02i)13-s − 0.598·14-s + (0.879 + 0.475i)15-s + 0.651·16-s + (−0.782 − 0.782i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.498i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.866 + 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $-0.866 + 0.498i$
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: \((2,\ 15,\ (\ :9/2),\ -0.866 + 0.498i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.175803 - 0.658482i\)
\(L(\frac12)\) \(\approx\) \(0.175803 - 0.658482i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-21.4 - 138. i)T \)
5 \( 1 + (-844. + 1.11e3i)T \)
good2 \( 1 + (24.1 + 24.1i)T + 512iT^{2} \)
7 \( 1 + (-1.78e3 + 1.78e3i)T - 4.03e7iT^{2} \)
11 \( 1 + 2.05e4iT - 2.35e9T^{2} \)
13 \( 1 + (1.05e5 + 1.05e5i)T + 1.06e10iT^{2} \)
17 \( 1 + (2.69e5 + 2.69e5i)T + 1.18e11iT^{2} \)
19 \( 1 + 7.79e5iT - 3.22e11T^{2} \)
23 \( 1 + (7.58e5 - 7.58e5i)T - 1.80e12iT^{2} \)
29 \( 1 - 3.30e6T + 1.45e13T^{2} \)
31 \( 1 - 5.40e6T + 2.64e13T^{2} \)
37 \( 1 + (9.21e6 - 9.21e6i)T - 1.29e14iT^{2} \)
41 \( 1 - 2.96e6iT - 3.27e14T^{2} \)
43 \( 1 + (2.26e7 + 2.26e7i)T + 5.02e14iT^{2} \)
47 \( 1 + (-2.16e7 - 2.16e7i)T + 1.11e15iT^{2} \)
53 \( 1 + (-5.81e7 + 5.81e7i)T - 3.29e15iT^{2} \)
59 \( 1 - 5.01e7T + 8.66e15T^{2} \)
61 \( 1 + 1.29e8T + 1.16e16T^{2} \)
67 \( 1 + (-4.93e7 + 4.93e7i)T - 2.72e16iT^{2} \)
71 \( 1 + 1.66e8iT - 4.58e16T^{2} \)
73 \( 1 + (-1.46e8 - 1.46e8i)T + 5.88e16iT^{2} \)
79 \( 1 - 2.00e7iT - 1.19e17T^{2} \)
83 \( 1 + (3.15e8 - 3.15e8i)T - 1.86e17iT^{2} \)
89 \( 1 - 3.21e8T + 3.50e17T^{2} \)
97 \( 1 + (1.74e8 - 1.74e8i)T - 7.60e17iT^{2} \)
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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

−17.12597903216181049560715945744, −15.56203446069722194200074550470, −13.68175207036207070878950552012, −11.84964449009970853659579636992, −10.48204039999099315991773267431, −9.499894478377803556408442743554, −8.385930468607201102057510289874, −4.99916388625302725367988532408, −2.64809933856417760551575611059, −0.47642119249524970829510633015, 1.96925725969678380237089420947, 6.15873595731142437542744719997, 7.16003836839926310324721996124, 8.542535016040082900586574334604, 10.07553750107262652591273860197, 12.17041378478070127078676122523, 14.10969190576423341157103086806, 15.00594663095885360535439923149, 16.85963613389077333832396040209, 17.76305988889276588001392947389

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