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Properties

Label 2-15-15.2-c17-0-25
Degree $2$
Conductor $15$
Sign $0.0242 + 0.999i$
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  + (258. − 258. i)2-s + (9.62e3 − 6.03e3i)3-s − 2.15e3i·4-s + (−4.04e5 − 7.74e5i)5-s + (9.27e5 − 4.04e6i)6-s + (1.90e7 + 1.90e7i)7-s + (3.32e7 + 3.32e7i)8-s + (5.62e7 − 1.16e8i)9-s + (−3.04e8 − 9.53e7i)10-s − 8.25e8i·11-s + (−1.29e7 − 2.07e7i)12-s + (2.29e9 − 2.29e9i)13-s + 9.81e9·14-s + (−8.56e9 − 5.01e9i)15-s + 1.74e10·16-s + (−8.41e9 + 8.41e9i)17-s + ⋯
L(s)  = 1  + (0.712 − 0.712i)2-s + (0.847 − 0.531i)3-s − 0.0164i·4-s + (−0.463 − 0.886i)5-s + (0.225 − 0.982i)6-s + (1.24 + 1.24i)7-s + (0.701 + 0.701i)8-s + (0.435 − 0.900i)9-s + (−0.961 − 0.301i)10-s − 1.16i·11-s + (−0.00871 − 0.0139i)12-s + (0.781 − 0.781i)13-s + 1.77·14-s + (−0.863 − 0.504i)15-s + 1.01·16-s + (−0.292 + 0.292i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0242 + 0.999i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.0242 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.0242 + 0.999i$
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: \((2,\ 15,\ (\ :17/2),\ 0.0242 + 0.999i)\)

Particular Values

\(L(9)\) \(\approx\) \(4.349930459\)
\(L(\frac12)\) \(\approx\) \(4.349930459\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-9.62e3 + 6.03e3i)T \)
5 \( 1 + (4.04e5 + 7.74e5i)T \)
good2 \( 1 + (-258. + 258. i)T - 1.31e5iT^{2} \)
7 \( 1 + (-1.90e7 - 1.90e7i)T + 2.32e14iT^{2} \)
11 \( 1 + 8.25e8iT - 5.05e17T^{2} \)
13 \( 1 + (-2.29e9 + 2.29e9i)T - 8.65e18iT^{2} \)
17 \( 1 + (8.41e9 - 8.41e9i)T - 8.27e20iT^{2} \)
19 \( 1 + 1.95e9iT - 5.48e21T^{2} \)
23 \( 1 + (2.21e11 + 2.21e11i)T + 1.41e23iT^{2} \)
29 \( 1 - 2.03e12T + 7.25e24T^{2} \)
31 \( 1 + 2.70e12T + 2.25e25T^{2} \)
37 \( 1 + (-9.34e12 - 9.34e12i)T + 4.56e26iT^{2} \)
41 \( 1 + 7.40e13iT - 2.61e27T^{2} \)
43 \( 1 + (2.79e13 - 2.79e13i)T - 5.87e27iT^{2} \)
47 \( 1 + (1.17e13 - 1.17e13i)T - 2.66e28iT^{2} \)
53 \( 1 + (1.80e14 + 1.80e14i)T + 2.05e29iT^{2} \)
59 \( 1 - 1.09e15T + 1.27e30T^{2} \)
61 \( 1 - 1.89e15T + 2.24e30T^{2} \)
67 \( 1 + (-2.18e15 - 2.18e15i)T + 1.10e31iT^{2} \)
71 \( 1 - 5.74e15iT - 2.96e31T^{2} \)
73 \( 1 + (6.58e15 - 6.58e15i)T - 4.74e31iT^{2} \)
79 \( 1 + 1.69e15iT - 1.81e32T^{2} \)
83 \( 1 + (-2.20e16 - 2.20e16i)T + 4.21e32iT^{2} \)
89 \( 1 + 5.65e16T + 1.37e33T^{2} \)
97 \( 1 + (5.79e16 + 5.79e16i)T + 5.95e33iT^{2} \)
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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

−14.48128003773980052864987362295, −13.25617679886705360717938297631, −12.26128967919724074839695471986, −11.27391309835261229694354865541, −8.456679563408699993095535684760, −8.290487928355187965020981247085, −5.45469113175846636700823132869, −3.88964445931507700418133011639, −2.47900518874711790337473528440, −1.19181884733000310449183881989, 1.68939363122647288669345682798, 3.89909688738057020631073270743, 4.63911339980067012244694492287, 6.93431442807191635319998577099, 7.84573239118344078921397410861, 10.01473802515382022465703604965, 11.13327629624531257598848830395, 13.60373786645343760094523697522, 14.35645714307963019158647050047, 15.10363039855025553779158552355

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