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Properties

Label 2-15-5.3-c4-0-1
Degree $2$
Conductor $15$
Sign $0.112 - 0.993i$
Analytic cond. $1.55054$
Root an. cond. $1.24521$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.08 + 2.08i)2-s + (3.67 + 3.67i)3-s + 7.34i·4-s + (−8.43 + 23.5i)5-s − 15.2·6-s + (65.1 − 65.1i)7-s + (−48.5 − 48.5i)8-s + 27i·9-s + (−31.4 − 66.5i)10-s + 56.3·11-s + (−26.9 + 26.9i)12-s + (0.983 + 0.983i)13-s + 270. i·14-s + (−117. + 55.4i)15-s + 84.5·16-s + (159. − 159. i)17-s + ⋯
L(s)  = 1  + (−0.520 + 0.520i)2-s + (0.408 + 0.408i)3-s + 0.458i·4-s + (−0.337 + 0.941i)5-s − 0.424·6-s + (1.32 − 1.32i)7-s + (−0.758 − 0.758i)8-s + 0.333i·9-s + (−0.314 − 0.665i)10-s + 0.466·11-s + (−0.187 + 0.187i)12-s + (0.00582 + 0.00582i)13-s + 1.38i·14-s + (−0.522 + 0.246i)15-s + 0.330·16-s + (0.550 − 0.550i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.112 - 0.993i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.112 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.112 - 0.993i$
Analytic conductor: \(1.55054\)
Root analytic conductor: \(1.24521\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :2),\ 0.112 - 0.993i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.773033 + 0.690767i\)
\(L(\frac12)\) \(\approx\) \(0.773033 + 0.690767i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-3.67 - 3.67i)T \)
5 \( 1 + (8.43 - 23.5i)T \)
good2 \( 1 + (2.08 - 2.08i)T - 16iT^{2} \)
7 \( 1 + (-65.1 + 65.1i)T - 2.40e3iT^{2} \)
11 \( 1 - 56.3T + 1.46e4T^{2} \)
13 \( 1 + (-0.983 - 0.983i)T + 2.85e4iT^{2} \)
17 \( 1 + (-159. + 159. i)T - 8.35e4iT^{2} \)
19 \( 1 - 265. iT - 1.30e5T^{2} \)
23 \( 1 + (185. + 185. i)T + 2.79e5iT^{2} \)
29 \( 1 + 544. iT - 7.07e5T^{2} \)
31 \( 1 + 710.T + 9.23e5T^{2} \)
37 \( 1 + (639. - 639. i)T - 1.87e6iT^{2} \)
41 \( 1 + 1.32e3T + 2.82e6T^{2} \)
43 \( 1 + (22.3 + 22.3i)T + 3.41e6iT^{2} \)
47 \( 1 + (-456. + 456. i)T - 4.87e6iT^{2} \)
53 \( 1 + (424. + 424. i)T + 7.89e6iT^{2} \)
59 \( 1 - 3.46e3iT - 1.21e7T^{2} \)
61 \( 1 - 4.93e3T + 1.38e7T^{2} \)
67 \( 1 + (3.80e3 - 3.80e3i)T - 2.01e7iT^{2} \)
71 \( 1 - 7.95e3T + 2.54e7T^{2} \)
73 \( 1 + (1.94e3 + 1.94e3i)T + 2.83e7iT^{2} \)
79 \( 1 - 4.08e3iT - 3.89e7T^{2} \)
83 \( 1 + (9.10e3 + 9.10e3i)T + 4.74e7iT^{2} \)
89 \( 1 - 7.01e3iT - 6.27e7T^{2} \)
97 \( 1 + (-2.57e3 + 2.57e3i)T - 8.85e7iT^{2} \)
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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

−18.64348454914237479934499292337, −17.48921789092751526822064674624, −16.43005607477516101637792741124, −14.89964633872149849278697181704, −13.94344395282377809556926847972, −11.62546862493771462214873354531, −10.17792517255598073271382228798, −8.176273455016128514592519486939, −7.18272803020851870699422545652, −3.87283545399608238211153861663, 1.68035869254979326225587030828, 5.36473749132794093293437779109, 8.295989096159804705762791380459, 9.185104374136714160875130453939, 11.35540306174801438388166646469, 12.36013610192514077518251305569, 14.37330894565591225522574638448, 15.41698655654696246711381111598, 17.44810255768590886003972278022, 18.49228985114559023003074496801

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