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Properties

Label 2-15-15.8-c17-0-16
Degree $2$
Conductor $15$
Sign $0.526 + 0.850i$
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  + (103. + 103. i)2-s + (−1.10e4 − 2.61e3i)3-s − 1.09e5i·4-s + (−2.00e5 + 8.50e5i)5-s + (−8.70e5 − 1.40e6i)6-s + (−1.46e7 + 1.46e7i)7-s + (2.48e7 − 2.48e7i)8-s + (1.15e8 + 5.77e7i)9-s + (−1.08e8 + 6.69e7i)10-s + 4.70e8i·11-s + (−2.86e8 + 1.21e9i)12-s + (1.86e7 + 1.86e7i)13-s − 3.02e9·14-s + (4.43e9 − 8.88e9i)15-s − 9.28e9·16-s + (−1.49e10 − 1.49e10i)17-s + ⋯
L(s)  = 1  + (0.284 + 0.284i)2-s + (−0.973 − 0.229i)3-s − 0.838i·4-s + (−0.229 + 0.973i)5-s + (−0.211 − 0.342i)6-s + (−0.963 + 0.963i)7-s + (0.523 − 0.523i)8-s + (0.894 + 0.447i)9-s + (−0.342 + 0.211i)10-s + 0.661i·11-s + (−0.192 + 0.815i)12-s + (0.00633 + 0.00633i)13-s − 0.548·14-s + (0.446 − 0.894i)15-s − 0.540·16-s + (−0.520 − 0.520i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.526 + 0.850i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.526 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.526 + 0.850i$
Analytic conductor: \(27.4833\)
Root analytic conductor: \(5.24245\)
Motivic weight: \(17\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :17/2),\ 0.526 + 0.850i)\)

Particular Values

\(L(9)\) \(\approx\) \(0.9308530300\)
\(L(\frac12)\) \(\approx\) \(0.9308530300\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.10e4 + 2.61e3i)T \)
5 \( 1 + (2.00e5 - 8.50e5i)T \)
good2 \( 1 + (-103. - 103. i)T + 1.31e5iT^{2} \)
7 \( 1 + (1.46e7 - 1.46e7i)T - 2.32e14iT^{2} \)
11 \( 1 - 4.70e8iT - 5.05e17T^{2} \)
13 \( 1 + (-1.86e7 - 1.86e7i)T + 8.65e18iT^{2} \)
17 \( 1 + (1.49e10 + 1.49e10i)T + 8.27e20iT^{2} \)
19 \( 1 + 8.78e10iT - 5.48e21T^{2} \)
23 \( 1 + (-9.12e10 + 9.12e10i)T - 1.41e23iT^{2} \)
29 \( 1 - 3.20e12T + 7.25e24T^{2} \)
31 \( 1 - 7.46e12T + 2.25e25T^{2} \)
37 \( 1 + (-9.72e12 + 9.72e12i)T - 4.56e26iT^{2} \)
41 \( 1 + 5.05e13iT - 2.61e27T^{2} \)
43 \( 1 + (3.01e13 + 3.01e13i)T + 5.87e27iT^{2} \)
47 \( 1 + (3.00e13 + 3.00e13i)T + 2.66e28iT^{2} \)
53 \( 1 + (4.91e14 - 4.91e14i)T - 2.05e29iT^{2} \)
59 \( 1 - 1.68e15T + 1.27e30T^{2} \)
61 \( 1 + 1.85e15T + 2.24e30T^{2} \)
67 \( 1 + (-2.60e15 + 2.60e15i)T - 1.10e31iT^{2} \)
71 \( 1 + 8.97e15iT - 2.96e31T^{2} \)
73 \( 1 + (-4.66e15 - 4.66e15i)T + 4.74e31iT^{2} \)
79 \( 1 - 1.13e16iT - 1.81e32T^{2} \)
83 \( 1 + (-1.76e16 + 1.76e16i)T - 4.21e32iT^{2} \)
89 \( 1 - 2.81e16T + 1.37e33T^{2} \)
97 \( 1 + (3.02e16 - 3.02e16i)T - 5.95e33iT^{2} \)
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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

−15.32063617863807936159518518670, −13.62290673995714766504929726243, −12.14723244319308598559206795167, −10.81051768981883015008074883347, −9.628682808876666438044267546387, −6.92806078664777939748959638287, −6.23652377033029158530327229361, −4.76606780885496545997861089102, −2.44931014889513442416846193153, −0.39789448008119846234189924464, 0.978903989283452690573516606770, 3.55955671906485797940768543221, 4.61163228267299795766228131907, 6.43889818691576178424820273278, 8.164666837537414233257765096016, 10.01365174012546520356118684918, 11.49705427455221948695587549576, 12.61920635436190707295063647041, 13.45964646985308848961471116988, 16.00287862483472119138269501461

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