Properties

Degree 2
Conductor $ 3 \cdot 5 $
Sign $0.846 + 0.533i$
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 + (−2.80 + 4.37i)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 + (27.0 + 17.3i)12-s + (2.82 − 2.82i)13-s + 49.8·14-s + (52.1 − 25.5i)15-s + 75.2·16-s + (−64.2 + 64.2i)17-s + ⋯
L(s)  = 1  + (0.941 − 0.941i)2-s + (−0.539 + 0.841i)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 + (0.650 + 0.416i)12-s + (0.0601 − 0.0601i)13-s + 0.951·14-s + (0.898 − 0.439i)15-s + 1.17·16-s + (−0.916 + 0.916i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 + 0.533i)\, \overline{\Lambda}(4-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.846 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(15\)    =    \(3 \cdot 5\)
\( \varepsilon \)  =  $0.846 + 0.533i$
motivic weight  =  \(3\)
character  :  $\chi_{15} (2, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 15,\ (\ :3/2),\ 0.846 + 0.533i)$
$L(2)$  $\approx$  $1.18094 - 0.340966i$
$L(\frac12)$  $\approx$  $1.18094 - 0.340966i$
$L(\frac{5}{2})$   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 \( 1 + (2.80 - 4.37i)T \)
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

−19.30259958825441787742596153992, −17.31568500594162903876429467897, −15.94156741440297609347277869749, −14.72388457287868043629627433549, −12.92467499929046952299674216287, −11.63044493892586629454532156164, −10.92675774076997381220493059372, −8.643475881773764191144968347928, −5.29863128763724854429367332798, −3.81923699294316135353858380830, 4.69659936412278108134805905703, 6.74591623147387126665616869644, 7.64039622096538033022724280243, 10.92667738166084842657161902615, 12.38248403242878978876885405111, 13.75551485231863316823742948562, 14.90062071875674611129003865106, 16.16625965304999888363950776273, 17.54097775571835031410946009303, 18.85738092280999553658893370885

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