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Properties

Label 2-15-15.8-c3-0-1
Degree $2$
Conductor $15$
Sign $0.846 - 0.533i$
Analytic cond. $0.885028$
Root an. cond. $0.940759$
Motivic weight $3$
Arithmetic yes
Rational no
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

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.846 - 0.533i$
Analytic conductor: \(0.885028\)
Root analytic conductor: \(0.940759\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :3/2),\ 0.846 - 0.533i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.18094 + 0.340966i\)
\(L(\frac12)\) \(\approx\) \(1.18094 + 0.340966i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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