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Properties

Label 2-15-5.3-c4-0-2
Degree $2$
Conductor $15$
Sign $0.806 + 0.590i$
Analytic cond. $1.55054$
Root an. cond. $1.24521$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.30 − 3.30i)2-s + (3.67 + 3.67i)3-s − 5.84i·4-s + (−16.2 − 19.0i)5-s + 24.2·6-s + (−33.1 + 33.1i)7-s + (33.5 + 33.5i)8-s + 27i·9-s + (−116. − 9.14i)10-s + 55.3·11-s + (21.4 − 21.4i)12-s + (−161. − 161. i)13-s + 219. i·14-s + (10.1 − 129. i)15-s + 315.·16-s + (278. − 278. i)17-s + ⋯
L(s)  = 1  + (0.826 − 0.826i)2-s + (0.408 + 0.408i)3-s − 0.365i·4-s + (−0.649 − 0.760i)5-s + 0.674·6-s + (−0.676 + 0.676i)7-s + (0.524 + 0.524i)8-s + 0.333i·9-s + (−1.16 − 0.0914i)10-s + 0.457·11-s + (0.149 − 0.149i)12-s + (−0.958 − 0.958i)13-s + 1.11i·14-s + (0.0451 − 0.575i)15-s + 1.23·16-s + (0.965 − 0.965i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 + 0.590i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.806 + 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.806 + 0.590i$
Analytic conductor: \(1.55054\)
Root analytic conductor: \(1.24521\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :2),\ 0.806 + 0.590i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.58798 - 0.519111i\)
\(L(\frac12)\) \(\approx\) \(1.58798 - 0.519111i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-3.67 - 3.67i)T \)
5 \( 1 + (16.2 + 19.0i)T \)
good2 \( 1 + (-3.30 + 3.30i)T - 16iT^{2} \)
7 \( 1 + (33.1 - 33.1i)T - 2.40e3iT^{2} \)
11 \( 1 - 55.3T + 1.46e4T^{2} \)
13 \( 1 + (161. + 161. i)T + 2.85e4iT^{2} \)
17 \( 1 + (-278. + 278. i)T - 8.35e4iT^{2} \)
19 \( 1 - 179. iT - 1.30e5T^{2} \)
23 \( 1 + (398. + 398. i)T + 2.79e5iT^{2} \)
29 \( 1 - 547. iT - 7.07e5T^{2} \)
31 \( 1 - 1.53e3T + 9.23e5T^{2} \)
37 \( 1 + (1.66e3 - 1.66e3i)T - 1.87e6iT^{2} \)
41 \( 1 + 307.T + 2.82e6T^{2} \)
43 \( 1 + (-104. - 104. i)T + 3.41e6iT^{2} \)
47 \( 1 + (346. - 346. i)T - 4.87e6iT^{2} \)
53 \( 1 + (2.02e3 + 2.02e3i)T + 7.89e6iT^{2} \)
59 \( 1 + 2.85e3iT - 1.21e7T^{2} \)
61 \( 1 + 1.05e3T + 1.38e7T^{2} \)
67 \( 1 + (-3.75e3 + 3.75e3i)T - 2.01e7iT^{2} \)
71 \( 1 - 1.42e3T + 2.54e7T^{2} \)
73 \( 1 + (-813. - 813. i)T + 2.83e7iT^{2} \)
79 \( 1 - 4.85e3iT - 3.89e7T^{2} \)
83 \( 1 + (1.31e3 + 1.31e3i)T + 4.74e7iT^{2} \)
89 \( 1 + 5.18e3iT - 6.27e7T^{2} \)
97 \( 1 + (-3.49e3 + 3.49e3i)T - 8.85e7iT^{2} \)
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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

−19.09727594910672482444290947577, −16.89201874548775812563848385073, −15.65017255344607472505777360897, −14.19541307119015917975804875859, −12.60677120314809891740787812574, −11.93803610537411434222863424142, −9.926453737509941217196043438808, −8.144737034280359298080560943850, −4.97046087383147559606186351794, −3.17127249986304817298574641530, 3.89992702479605424022268000691, 6.50061281637004492324347333038, 7.53879452563094108768082697690, 10.05580602450662139506037929379, 12.13751659192174687118146847774, 13.73815605190543244182975527935, 14.56231748222562017787141320707, 15.73386057194929203711414668198, 17.09626107865314265210419991985, 19.22558887696408618681959449132

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