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Properties

Label 2-75-15.2-c9-0-13
Degree $2$
Conductor $75$
Sign $0.144 - 0.989i$
Analytic cond. $38.6276$
Root an. cond. $6.21511$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−13.8 + 13.8i)2-s + (128. − 55.7i)3-s + 127. i·4-s + (−1.01e3 + 2.55e3i)6-s + (−6.79e3 − 6.79e3i)7-s + (−8.86e3 − 8.86e3i)8-s + (1.34e4 − 1.43e4i)9-s + 2.84e4i·11-s + (7.07e3 + 1.63e4i)12-s + (−2.82e4 + 2.82e4i)13-s + 1.88e5·14-s + 1.80e5·16-s + (2.11e4 − 2.11e4i)17-s + (1.21e4 + 3.85e5i)18-s + 7.59e5i·19-s + ⋯
L(s)  = 1  + (−0.613 + 0.613i)2-s + (0.917 − 0.397i)3-s + 0.248i·4-s + (−0.319 + 0.806i)6-s + (−1.07 − 1.07i)7-s + (−0.765 − 0.765i)8-s + (0.684 − 0.728i)9-s + 0.584i·11-s + (0.0985 + 0.227i)12-s + (−0.273 + 0.273i)13-s + 1.31·14-s + 0.690·16-s + (0.0612 − 0.0612i)17-s + (0.0272 + 0.866i)18-s + 1.33i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.144 - 0.989i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.144 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.144 - 0.989i$
Analytic conductor: \(38.6276\)
Root analytic conductor: \(6.21511\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :9/2),\ 0.144 - 0.989i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.12449 + 0.972071i\)
\(L(\frac12)\) \(\approx\) \(1.12449 + 0.972071i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-128. + 55.7i)T \)
5 \( 1 \)
good2 \( 1 + (13.8 - 13.8i)T - 512iT^{2} \)
7 \( 1 + (6.79e3 + 6.79e3i)T + 4.03e7iT^{2} \)
11 \( 1 - 2.84e4iT - 2.35e9T^{2} \)
13 \( 1 + (2.82e4 - 2.82e4i)T - 1.06e10iT^{2} \)
17 \( 1 + (-2.11e4 + 2.11e4i)T - 1.18e11iT^{2} \)
19 \( 1 - 7.59e5iT - 3.22e11T^{2} \)
23 \( 1 + (-1.18e6 - 1.18e6i)T + 1.80e12iT^{2} \)
29 \( 1 - 6.93e6T + 1.45e13T^{2} \)
31 \( 1 + 1.45e6T + 2.64e13T^{2} \)
37 \( 1 + (-8.18e6 - 8.18e6i)T + 1.29e14iT^{2} \)
41 \( 1 + 7.44e6iT - 3.27e14T^{2} \)
43 \( 1 + (6.21e6 - 6.21e6i)T - 5.02e14iT^{2} \)
47 \( 1 + (1.95e7 - 1.95e7i)T - 1.11e15iT^{2} \)
53 \( 1 + (-4.55e7 - 4.55e7i)T + 3.29e15iT^{2} \)
59 \( 1 - 3.03e7T + 8.66e15T^{2} \)
61 \( 1 + 5.71e7T + 1.16e16T^{2} \)
67 \( 1 + (-1.30e8 - 1.30e8i)T + 2.72e16iT^{2} \)
71 \( 1 - 1.62e8iT - 4.58e16T^{2} \)
73 \( 1 + (-1.40e8 + 1.40e8i)T - 5.88e16iT^{2} \)
79 \( 1 - 4.65e7iT - 1.19e17T^{2} \)
83 \( 1 + (-4.75e8 - 4.75e8i)T + 1.86e17iT^{2} \)
89 \( 1 + 1.99e8T + 3.50e17T^{2} \)
97 \( 1 + (9.83e8 + 9.83e8i)T + 7.60e17iT^{2} \)
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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

−12.99423543487160174017615502019, −12.17988436701585917426384956328, −10.09092247820635377306535820386, −9.439073976251205294996057745771, −8.159995305159647907481761993330, −7.24049398167216197202656610380, −6.52090508861614003378518096108, −4.01802625572704447655575457843, −2.98244583220014939317935736280, −1.03401315331440710204164439561, 0.56411621864865160956698596123, 2.40837530455399379907893655486, 3.06338213937642005893977194661, 5.07869349863316626332022304576, 6.54463945103230097792837059513, 8.456637484824093236014553675760, 9.115372244289470581404956862164, 9.962393814416293056060692595671, 11.00463428750362293349507961009, 12.36065983955462532199708908439

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