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Properties

Label 2-15-15.2-c17-0-5
Degree $2$
Conductor $15$
Sign $0.0798 + 0.996i$
Analytic cond. $27.4833$
Root an. cond. $5.24245$
Motivic weight $17$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−493. + 493. i)2-s + (−9.45e3 + 6.31e3i)3-s − 3.55e5i·4-s + (4.25e5 + 7.62e5i)5-s + (1.54e6 − 7.77e6i)6-s + (7.63e6 + 7.63e6i)7-s + (1.10e8 + 1.10e8i)8-s + (4.94e7 − 1.19e8i)9-s + (−5.85e8 − 1.66e8i)10-s + 1.58e8i·11-s + (2.24e9 + 3.35e9i)12-s + (−2.74e9 + 2.74e9i)13-s − 7.52e9·14-s + (−8.83e9 − 4.52e9i)15-s − 6.24e10·16-s + (5.83e8 − 5.83e8i)17-s + ⋯
L(s)  = 1  + (−1.36 + 1.36i)2-s + (−0.831 + 0.555i)3-s − 2.71i·4-s + (0.486 + 0.873i)5-s + (0.376 − 1.88i)6-s + (0.500 + 0.500i)7-s + (2.32 + 2.32i)8-s + (0.383 − 0.923i)9-s + (−1.85 − 0.526i)10-s + 0.223i·11-s + (1.50 + 2.25i)12-s + (−0.931 + 0.931i)13-s − 1.36·14-s + (−0.889 − 0.456i)15-s − 3.63·16-s + (0.0202 − 0.0202i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0798 + 0.996i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.0798 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.0798 + 0.996i$
Analytic conductor: \(27.4833\)
Root analytic conductor: \(5.24245\)
Motivic weight: \(17\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :17/2),\ 0.0798 + 0.996i)\)

Particular Values

\(L(9)\) \(\approx\) \(0.4823201415\)
\(L(\frac12)\) \(\approx\) \(0.4823201415\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (9.45e3 - 6.31e3i)T \)
5 \( 1 + (-4.25e5 - 7.62e5i)T \)
good2 \( 1 + (493. - 493. i)T - 1.31e5iT^{2} \)
7 \( 1 + (-7.63e6 - 7.63e6i)T + 2.32e14iT^{2} \)
11 \( 1 - 1.58e8iT - 5.05e17T^{2} \)
13 \( 1 + (2.74e9 - 2.74e9i)T - 8.65e18iT^{2} \)
17 \( 1 + (-5.83e8 + 5.83e8i)T - 8.27e20iT^{2} \)
19 \( 1 - 8.96e10iT - 5.48e21T^{2} \)
23 \( 1 + (-3.76e11 - 3.76e11i)T + 1.41e23iT^{2} \)
29 \( 1 - 9.43e11T + 7.25e24T^{2} \)
31 \( 1 + 2.34e12T + 2.25e25T^{2} \)
37 \( 1 + (-9.66e11 - 9.66e11i)T + 4.56e26iT^{2} \)
41 \( 1 - 4.75e13iT - 2.61e27T^{2} \)
43 \( 1 + (-4.22e13 + 4.22e13i)T - 5.87e27iT^{2} \)
47 \( 1 + (1.75e14 - 1.75e14i)T - 2.66e28iT^{2} \)
53 \( 1 + (-8.24e12 - 8.24e12i)T + 2.05e29iT^{2} \)
59 \( 1 - 1.62e15T + 1.27e30T^{2} \)
61 \( 1 + 6.24e14T + 2.24e30T^{2} \)
67 \( 1 + (4.35e15 + 4.35e15i)T + 1.10e31iT^{2} \)
71 \( 1 - 2.04e15iT - 2.96e31T^{2} \)
73 \( 1 + (-5.28e14 + 5.28e14i)T - 4.74e31iT^{2} \)
79 \( 1 + 6.60e15iT - 1.81e32T^{2} \)
83 \( 1 + (3.27e15 + 3.27e15i)T + 4.21e32iT^{2} \)
89 \( 1 - 1.15e16T + 1.37e33T^{2} \)
97 \( 1 + (-4.31e16 - 4.31e16i)T + 5.95e33iT^{2} \)
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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

−16.46001260368266845384268305365, −15.12830247623754895321810434111, −14.44434393532800430795644348798, −11.41648670148870373132904905788, −10.15207722246484565697740985659, −9.257985140103152378069929934998, −7.39157961624364498183398621219, −6.24426391987156695450056418423, −5.10156547296120915356233885287, −1.61986585268014128229151944278, 0.34050556701342350248236092107, 1.04964029223922869506081276681, 2.42042337768168084319245327710, 4.78389673981859593760362455241, 7.31989599372054527722550806444, 8.628240414528457349229156435573, 10.13147678178718807239910317700, 11.16062038887331650998915285551, 12.41609097233555577218195704726, 13.23179754737090556411614649806

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