Properties

Degree 2
Conductor $ 3 \cdot 5 $
Sign $0.817 - 0.575i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.18 + 1.18i)2-s + (−0.932 + 5.11i)3-s − 5.17i·4-s + (2.48 − 10.9i)5-s + (−7.17 + 4.96i)6-s + (−13.3 + 13.3i)7-s + (15.6 − 15.6i)8-s + (−25.2 − 9.53i)9-s + (15.8 − 10i)10-s + 28.7i·11-s + (26.4 + 4.83i)12-s + (14.1 + 14.1i)13-s − 31.7·14-s + (53.4 + 22.8i)15-s − 4.25·16-s + (18.5 + 18.5i)17-s + ⋯
L(s)  = 1  + (0.419 + 0.419i)2-s + (−0.179 + 0.983i)3-s − 0.647i·4-s + (0.221 − 0.975i)5-s + (−0.488 + 0.337i)6-s + (−0.721 + 0.721i)7-s + (0.691 − 0.691i)8-s + (−0.935 − 0.353i)9-s + (0.502 − 0.316i)10-s + 0.787i·11-s + (0.636 + 0.116i)12-s + (0.302 + 0.302i)13-s − 0.605·14-s + (0.919 + 0.393i)15-s − 0.0664·16-s + (0.264 + 0.264i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.817 - 0.575i)\, \overline{\Lambda}(4-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.817 - 0.575i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(15\)    =    \(3 \cdot 5\)
\( \varepsilon \)  =  $0.817 - 0.575i$
motivic weight  =  \(3\)
character  :  $\chi_{15} (8, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 15,\ (\ :3/2),\ 0.817 - 0.575i)$
$L(2)$  $\approx$  $1.06255 + 0.336211i$
$L(\frac12)$  $\approx$  $1.06255 + 0.336211i$
$L(\frac{5}{2})$   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 \( 1 + (0.932 - 5.11i)T \)
5 \( 1 + (-2.48 + 10.9i)T \)
good2 \( 1 + (-1.18 - 1.18i)T + 8iT^{2} \)
7 \( 1 + (13.3 - 13.3i)T - 343iT^{2} \)
11 \( 1 - 28.7iT - 1.33e3T^{2} \)
13 \( 1 + (-14.1 - 14.1i)T + 2.19e3iT^{2} \)
17 \( 1 + (-18.5 - 18.5i)T + 4.91e3iT^{2} \)
19 \( 1 + 49.0iT - 6.85e3T^{2} \)
23 \( 1 + (37.7 - 37.7i)T - 1.21e4iT^{2} \)
29 \( 1 - 125.T + 2.43e4T^{2} \)
31 \( 1 - 247.T + 2.97e4T^{2} \)
37 \( 1 + (127. - 127. i)T - 5.06e4iT^{2} \)
41 \( 1 + 390. iT - 6.89e4T^{2} \)
43 \( 1 + (39.3 + 39.3i)T + 7.95e4iT^{2} \)
47 \( 1 + (-124. - 124. i)T + 1.03e5iT^{2} \)
53 \( 1 + (160. - 160. i)T - 1.48e5iT^{2} \)
59 \( 1 + 729.T + 2.05e5T^{2} \)
61 \( 1 - 2T + 2.26e5T^{2} \)
67 \( 1 + (329. - 329. i)T - 3.00e5iT^{2} \)
71 \( 1 - 171. iT - 3.57e5T^{2} \)
73 \( 1 + (279. + 279. i)T + 3.89e5iT^{2} \)
79 \( 1 + 48.0iT - 4.93e5T^{2} \)
83 \( 1 + (-144. + 144. i)T - 5.71e5iT^{2} \)
89 \( 1 - 1.41e3T + 7.04e5T^{2} \)
97 \( 1 + (-908. + 908. i)T - 9.12e5iT^{2} \)
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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

−19.25758081563556739303967659974, −17.35454966791286494606656471278, −15.95870364388380401553473878136, −15.39310970949471731172267744761, −13.80488559212347596046056032992, −12.19125059967706059234033023708, −10.14731040324561577735039196201, −9.070432271025360323698373940548, −6.05628949291157793786288403308, −4.67402148160883643636680785694, 3.12561044923699290414261651363, 6.44580005751450636991095555443, 7.932727633344145449525832173730, 10.60146458940347499721459938807, 11.93363037914259511849740978990, 13.33203520106563024015881967218, 14.07130552583954130263336486016, 16.40493507111722138718837048110, 17.61275748436745568513159476047, 18.84010441718059650542488254178

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