Properties

Label 2-75-15.14-c12-0-68
Degree $2$
Conductor $75$
Sign $-0.997 + 0.0762i$
Analytic cond. $68.5495$
Root an. cond. $8.27946$
Motivic weight $12$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 91.7·2-s + (275. − 675i)3-s + 4.32e3·4-s + (2.52e4 − 6.19e4i)6-s − 4.02e4i·7-s + 2.12e4·8-s + (−3.79e5 − 3.71e5i)9-s − 1.16e6i·11-s + (1.19e6 − 2.92e6i)12-s + 1.28e6i·13-s − 3.69e6i·14-s − 1.57e7·16-s + 1.48e7·17-s + (−3.48e7 − 3.41e7i)18-s − 5.33e7·19-s + ⋯
L(s)  = 1  + 1.43·2-s + (0.377 − 0.925i)3-s + 1.05·4-s + (0.541 − 1.32i)6-s − 0.342i·7-s + 0.0812·8-s + (−0.714 − 0.699i)9-s − 0.655i·11-s + (0.399 − 0.978i)12-s + 0.266i·13-s − 0.490i·14-s − 0.940·16-s + 0.614·17-s + (−1.02 − 1.00i)18-s − 1.13·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0762i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.997 + 0.0762i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.997 + 0.0762i$
Analytic conductor: \(68.5495\)
Root analytic conductor: \(8.27946\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :6),\ -0.997 + 0.0762i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(2.728060175\)
\(L(\frac12)\) \(\approx\) \(2.728060175\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-275. + 675i)T \)
5 \( 1 \)
good2 \( 1 - 91.7T + 4.09e3T^{2} \)
7 \( 1 + 4.02e4iT - 1.38e10T^{2} \)
11 \( 1 + 1.16e6iT - 3.13e12T^{2} \)
13 \( 1 - 1.28e6iT - 2.32e13T^{2} \)
17 \( 1 - 1.48e7T + 5.82e14T^{2} \)
19 \( 1 + 5.33e7T + 2.21e15T^{2} \)
23 \( 1 + 1.07e8T + 2.19e16T^{2} \)
29 \( 1 + 1.20e8iT - 3.53e17T^{2} \)
31 \( 1 - 6.65e7T + 7.87e17T^{2} \)
37 \( 1 + 2.22e9iT - 6.58e18T^{2} \)
41 \( 1 + 8.21e9iT - 2.25e19T^{2} \)
43 \( 1 - 8.97e9iT - 3.99e19T^{2} \)
47 \( 1 + 1.07e9T + 1.16e20T^{2} \)
53 \( 1 + 4.11e10T + 4.91e20T^{2} \)
59 \( 1 - 4.61e10iT - 1.77e21T^{2} \)
61 \( 1 + 4.06e10T + 2.65e21T^{2} \)
67 \( 1 + 1.21e11iT - 8.18e21T^{2} \)
71 \( 1 - 4.48e10iT - 1.64e22T^{2} \)
73 \( 1 + 6.09e10iT - 2.29e22T^{2} \)
79 \( 1 - 2.52e11T + 5.90e22T^{2} \)
83 \( 1 - 4.10e11T + 1.06e23T^{2} \)
89 \( 1 + 1.12e11iT - 2.46e23T^{2} \)
97 \( 1 + 6.53e11iT - 6.93e23T^{2} \)
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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

−12.05076982509235737854881061928, −10.89730042665858543491526226055, −9.088216396396218699373189317879, −7.82475036763747860740529747104, −6.54829216258847855452420280808, −5.72695316724097913707361684797, −4.18651447283822811675516337966, −3.13385180473722374498553261487, −1.93003837311573836504831850566, −0.33209265521488702333430184545, 2.18127333970212149685494714757, 3.28976115512873308701559259298, 4.33623556242560192959442615579, 5.21361284565541345576224283311, 6.35576900169028651753722948259, 8.099729580180423926136497282373, 9.388043745773122811593874044292, 10.53207615572527146574022716675, 11.77716357845930460366572011214, 12.72192578929696107101940118556

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