Properties

Degree 2
Conductor $ 3 \cdot 5 $
Sign $0.827 + 0.562i$
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 + (5.11 − 0.932i)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 + (−4.83 − 26.4i)12-s + (14.1 + 14.1i)13-s + 31.7·14-s + (−2.51 + 58.0i)15-s − 4.25·16-s + (−18.5 − 18.5i)17-s + ⋯
L(s)  = 1  + (−0.419 − 0.419i)2-s + (0.983 − 0.179i)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 + (−0.116 − 0.636i)12-s + (0.302 + 0.302i)13-s + 0.605·14-s + (−0.0433 + 0.999i)15-s − 0.0664·16-s + (−0.264 − 0.264i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.562i)\, \overline{\Lambda}(4-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.827 + 0.562i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(15\)    =    \(3 \cdot 5\)
\( \varepsilon \)  =  $0.827 + 0.562i$
motivic weight  =  \(3\)
character  :  $\chi_{15} (8, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 15,\ (\ :3/2),\ 0.827 + 0.562i)$
$L(2)$  $\approx$  $0.933936 - 0.287312i$
$L(\frac12)$  $\approx$  $0.933936 - 0.287312i$
$L(\frac{5}{2})$   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 \( 1 + (-5.11 + 0.932i)T \)
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

−18.98606645131647495282953597326, −18.22344697396416525847690746252, −15.73209268883751784536857699872, −14.77626857759741053425400027435, −13.51618850146190279934183483122, −11.49630306096740907800884784680, −9.964144928199166960579502353493, −8.688393218705620137484404251792, −6.50255990905518713887888240877, −2.82304810281741027661711502846, 3.90055542462926303190335709383, 7.30324020632429952807597508995, 8.576976398107080362336417453962, 9.832573702229850873819797035896, 12.49743549338424276033983434725, 13.43054321387428737362085751487, 15.39563343738707901094380443663, 16.33759711317536899547976342217, 17.47615302026504461901797489048, 19.19300339549821438778472856279

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