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Properties

Label 2-15-5.3-c10-0-9
Degree $2$
Conductor $15$
Sign $-0.922 + 0.385i$
Analytic cond. $9.53035$
Root an. cond. $3.08712$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (42.4 − 42.4i)2-s + (−99.2 − 99.2i)3-s − 2.58e3i·4-s + (3.08e3 − 511. i)5-s − 8.42e3·6-s + (−3.58e3 + 3.58e3i)7-s + (−6.61e4 − 6.61e4i)8-s + 1.96e4i·9-s + (1.09e5 − 1.52e5i)10-s − 1.62e5·11-s + (−2.56e5 + 2.56e5i)12-s + (3.95e5 + 3.95e5i)13-s + 3.04e5i·14-s + (−3.56e5 − 2.55e5i)15-s − 2.97e6·16-s + (1.74e6 − 1.74e6i)17-s + ⋯
L(s)  = 1  + (1.32 − 1.32i)2-s + (−0.408 − 0.408i)3-s − 2.52i·4-s + (0.986 − 0.163i)5-s − 1.08·6-s + (−0.213 + 0.213i)7-s + (−2.01 − 2.01i)8-s + 0.333i·9-s + (1.09 − 1.52i)10-s − 1.01·11-s + (−1.02 + 1.02i)12-s + (1.06 + 1.06i)13-s + 0.565i·14-s + (−0.469 − 0.335i)15-s − 2.83·16-s + (1.23 − 1.23i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.922 + 0.385i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.922 + 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $-0.922 + 0.385i$
Analytic conductor: \(9.53035\)
Root analytic conductor: \(3.08712\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :5),\ -0.922 + 0.385i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.631964 - 3.14910i\)
\(L(\frac12)\) \(\approx\) \(0.631964 - 3.14910i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (99.2 + 99.2i)T \)
5 \( 1 + (-3.08e3 + 511. i)T \)
good2 \( 1 + (-42.4 + 42.4i)T - 1.02e3iT^{2} \)
7 \( 1 + (3.58e3 - 3.58e3i)T - 2.82e8iT^{2} \)
11 \( 1 + 1.62e5T + 2.59e10T^{2} \)
13 \( 1 + (-3.95e5 - 3.95e5i)T + 1.37e11iT^{2} \)
17 \( 1 + (-1.74e6 + 1.74e6i)T - 2.01e12iT^{2} \)
19 \( 1 + 8.26e5iT - 6.13e12T^{2} \)
23 \( 1 + (8.16e5 + 8.16e5i)T + 4.14e13iT^{2} \)
29 \( 1 - 4.13e6iT - 4.20e14T^{2} \)
31 \( 1 - 2.32e7T + 8.19e14T^{2} \)
37 \( 1 + (3.90e7 - 3.90e7i)T - 4.80e15iT^{2} \)
41 \( 1 - 1.24e8T + 1.34e16T^{2} \)
43 \( 1 + (2.03e6 + 2.03e6i)T + 2.16e16iT^{2} \)
47 \( 1 + (2.81e8 - 2.81e8i)T - 5.25e16iT^{2} \)
53 \( 1 + (1.42e8 + 1.42e8i)T + 1.74e17iT^{2} \)
59 \( 1 - 2.82e8iT - 5.11e17T^{2} \)
61 \( 1 + 7.06e8T + 7.13e17T^{2} \)
67 \( 1 + (7.04e8 - 7.04e8i)T - 1.82e18iT^{2} \)
71 \( 1 - 2.19e9T + 3.25e18T^{2} \)
73 \( 1 + (2.52e8 + 2.52e8i)T + 4.29e18iT^{2} \)
79 \( 1 + 1.09e7iT - 9.46e18T^{2} \)
83 \( 1 + (-4.24e9 - 4.24e9i)T + 1.55e19iT^{2} \)
89 \( 1 + 1.52e9iT - 3.11e19T^{2} \)
97 \( 1 + (6.39e9 - 6.39e9i)T - 7.37e19iT^{2} \)
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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

−16.07965252181611137823720433664, −14.13273327573179400196904567510, −13.38486977116948067390709227931, −12.25657940165386476126477644114, −10.97770015651816246483937101572, −9.637386540289557987111100057000, −6.18745678967241746983313685431, −4.97717450887506730905647489975, −2.74843828837837906699193245048, −1.22879639618475168156480110923, 3.42067296885042128783180082432, 5.36304723675410741575440956413, 6.18271425625571389204666902020, 8.057942548683219443040293181684, 10.41866426072261270324392794421, 12.62639278633441523912444722211, 13.54252612509133965186903940365, 14.84815043700347174110902785421, 15.94729875932375528073217153979, 17.01791207286439866022472501413

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