Properties

Label 2-15-15.8-c9-0-5
Degree $2$
Conductor $15$
Sign $0.754 + 0.656i$
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  + (−20.3 − 20.3i)2-s + (−124. + 65.1i)3-s + 318. i·4-s + (−840. + 1.11e3i)5-s + (3.86e3 + 1.20e3i)6-s + (1.02e3 − 1.02e3i)7-s + (−3.93e3 + 3.93e3i)8-s + (1.11e4 − 1.61e4i)9-s + (3.98e4 − 5.62e3i)10-s − 8.23e4i·11-s + (−2.07e4 − 3.96e4i)12-s + (4.61e4 + 4.61e4i)13-s − 4.17e4·14-s + (3.16e4 − 1.93e5i)15-s + 3.23e5·16-s + (2.83e5 + 2.83e5i)17-s + ⋯
L(s)  = 1  + (−0.900 − 0.900i)2-s + (−0.885 + 0.464i)3-s + 0.622i·4-s + (−0.601 + 0.798i)5-s + (1.21 + 0.379i)6-s + (0.161 − 0.161i)7-s + (−0.339 + 0.339i)8-s + (0.568 − 0.822i)9-s + (1.26 − 0.177i)10-s − 1.69i·11-s + (−0.289 − 0.551i)12-s + (0.448 + 0.448i)13-s − 0.290·14-s + (0.161 − 0.986i)15-s + 1.23·16-s + (0.822 + 0.822i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.754 + 0.656i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.754 + 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.754 + 0.656i$
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: \((2,\ 15,\ (\ :9/2),\ 0.754 + 0.656i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.588489 - 0.220138i\)
\(L(\frac12)\) \(\approx\) \(0.588489 - 0.220138i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (124. - 65.1i)T \)
5 \( 1 + (840. - 1.11e3i)T \)
good2 \( 1 + (20.3 + 20.3i)T + 512iT^{2} \)
7 \( 1 + (-1.02e3 + 1.02e3i)T - 4.03e7iT^{2} \)
11 \( 1 + 8.23e4iT - 2.35e9T^{2} \)
13 \( 1 + (-4.61e4 - 4.61e4i)T + 1.06e10iT^{2} \)
17 \( 1 + (-2.83e5 - 2.83e5i)T + 1.18e11iT^{2} \)
19 \( 1 - 4.48e5iT - 3.22e11T^{2} \)
23 \( 1 + (-5.77e5 + 5.77e5i)T - 1.80e12iT^{2} \)
29 \( 1 - 3.17e6T + 1.45e13T^{2} \)
31 \( 1 - 4.84e6T + 2.64e13T^{2} \)
37 \( 1 + (2.00e6 - 2.00e6i)T - 1.29e14iT^{2} \)
41 \( 1 + 2.19e7iT - 3.27e14T^{2} \)
43 \( 1 + (-5.07e6 - 5.07e6i)T + 5.02e14iT^{2} \)
47 \( 1 + (1.80e7 + 1.80e7i)T + 1.11e15iT^{2} \)
53 \( 1 + (2.88e7 - 2.88e7i)T - 3.29e15iT^{2} \)
59 \( 1 - 1.79e8T + 8.66e15T^{2} \)
61 \( 1 + 2.05e7T + 1.16e16T^{2} \)
67 \( 1 + (-1.55e8 + 1.55e8i)T - 2.72e16iT^{2} \)
71 \( 1 - 3.49e8iT - 4.58e16T^{2} \)
73 \( 1 + (-5.01e7 - 5.01e7i)T + 5.88e16iT^{2} \)
79 \( 1 + 5.64e8iT - 1.19e17T^{2} \)
83 \( 1 + (-2.73e8 + 2.73e8i)T - 1.86e17iT^{2} \)
89 \( 1 - 2.28e8T + 3.50e17T^{2} \)
97 \( 1 + (-7.06e8 + 7.06e8i)T - 7.60e17iT^{2} \)
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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

−17.25534352680795679209010325470, −16.03151501301477741362351317011, −14.41865476117178792231968811962, −11.99976962731730224684458882728, −11.03977663506229182386711137374, −10.27184210874314477400173602516, −8.413243264940300335514642987377, −6.09044936568493353279843916248, −3.48266637584344454530217465056, −0.804452147388884140919898724632, 0.850252599368007062259335280320, 5.01455341315025717231243618708, 6.94658260844949202422737683986, 8.048392116945845869879989126467, 9.749973888885629937035588301770, 11.81427481006423351556424588199, 12.85828437442059906117613546742, 15.31960483799386961052040759277, 16.22563392942193842267674016256, 17.39724424671951237840020496850

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