Properties

Degree 2
Conductor $ 3 \cdot 5 $
Sign $-0.546 + 0.837i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.66 − 2.66i)2-s + (−4.37 − 2.80i)3-s + 6.17i·4-s + (9.55 − 5.80i)5-s + (4.17 + 19.1i)6-s + (9.35 − 9.35i)7-s + (−4.84 + 4.84i)8-s + (11.2 + 24.5i)9-s + (−40.8 − 10.0i)10-s − 34.1i·11-s + (17.3 − 27.0i)12-s + (2.82 + 2.82i)13-s − 49.8·14-s + (−58.0 − 1.42i)15-s + 75.2·16-s + (64.2 + 64.2i)17-s + ⋯
L(s)  = 1  + (−0.941 − 0.941i)2-s + (−0.841 − 0.539i)3-s + 0.772i·4-s + (0.854 − 0.518i)5-s + (0.284 + 1.30i)6-s + (0.505 − 0.505i)7-s + (−0.214 + 0.214i)8-s + (0.417 + 0.908i)9-s + (−1.29 − 0.316i)10-s − 0.935i·11-s + (0.416 − 0.650i)12-s + (0.0601 + 0.0601i)13-s − 0.951·14-s + (−0.999 − 0.0245i)15-s + 1.17·16-s + (0.916 + 0.916i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.546 + 0.837i)\, \overline{\Lambda}(4-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.546 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(15\)    =    \(3 \cdot 5\)
\( \varepsilon \)  =  $-0.546 + 0.837i$
motivic weight  =  \(3\)
character  :  $\chi_{15} (8, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 15,\ (\ :3/2),\ -0.546 + 0.837i)$
$L(2)$  $\approx$  $0.276037 - 0.509736i$
$L(\frac12)$  $\approx$  $0.276037 - 0.509736i$
$L(\frac{5}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{3,\;5\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 + (4.37 + 2.80i)T \)
5 \( 1 + (-9.55 + 5.80i)T \)
good2 \( 1 + (2.66 + 2.66i)T + 8iT^{2} \)
7 \( 1 + (-9.35 + 9.35i)T - 343iT^{2} \)
11 \( 1 + 34.1iT - 1.33e3T^{2} \)
13 \( 1 + (-2.82 - 2.82i)T + 2.19e3iT^{2} \)
17 \( 1 + (-64.2 - 64.2i)T + 4.91e3iT^{2} \)
19 \( 1 - 19.0iT - 6.85e3T^{2} \)
23 \( 1 + (51.4 - 51.4i)T - 1.21e4iT^{2} \)
29 \( 1 - 50.5T + 2.43e4T^{2} \)
31 \( 1 + 93.3T + 2.97e4T^{2} \)
37 \( 1 + (161. - 161. i)T - 5.06e4iT^{2} \)
41 \( 1 - 88.7iT - 6.89e4T^{2} \)
43 \( 1 + (-176. - 176. i)T + 7.95e4iT^{2} \)
47 \( 1 + (-38.2 - 38.2i)T + 1.03e5iT^{2} \)
53 \( 1 + (-344. + 344. i)T - 1.48e5iT^{2} \)
59 \( 1 + 421.T + 2.05e5T^{2} \)
61 \( 1 - 2T + 2.26e5T^{2} \)
67 \( 1 + (-430. + 430. i)T - 3.00e5iT^{2} \)
71 \( 1 - 733. iT - 3.57e5T^{2} \)
73 \( 1 + (348. + 348. i)T + 3.89e5iT^{2} \)
79 \( 1 - 588. iT - 4.93e5T^{2} \)
83 \( 1 + (217. - 217. i)T - 5.71e5iT^{2} \)
89 \( 1 + 1.27e3T + 7.04e5T^{2} \)
97 \( 1 + (432. - 432. i)T - 9.12e5iT^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−18.42252198082474334606652768980, −17.42267598750310190758412777532, −16.63601673806536230080255886795, −13.98511205091157486256486587168, −12.48898703138684231083296288898, −11.13755423713879435257530953752, −10.05023577054230860420308309352, −8.243528307960224561706635465053, −5.75433112901286944091839162579, −1.33574749407580803200854942549, 5.56961244628283330065767177089, 7.10783018415376297378217035178, 9.234104993362180615990450167760, 10.34632743298659957387878163491, 12.19605027495626958058919425381, 14.56316597896230275076176934785, 15.68360242232261921792185145524, 16.93798504706306063126105589936, 17.87571050321519678259714618329, 18.48415711277938393935036237044

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