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Properties

Label 2-75-5.3-c8-0-15
Degree $2$
Conductor $75$
Sign $0.991 - 0.130i$
Analytic cond. $30.5533$
Root an. cond. $5.52751$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−12.8 + 12.8i)2-s + (33.0 + 33.0i)3-s − 74.1i·4-s − 849.·6-s + (−333. + 333. i)7-s + (−2.33e3 − 2.33e3i)8-s + 2.18e3i·9-s + 9.14e3·11-s + (2.45e3 − 2.45e3i)12-s + (−1.23e4 − 1.23e4i)13-s − 8.55e3i·14-s + 7.90e4·16-s + (−7.86e3 + 7.86e3i)17-s + (−2.80e4 − 2.80e4i)18-s − 1.78e5i·19-s + ⋯
L(s)  = 1  + (−0.802 + 0.802i)2-s + (0.408 + 0.408i)3-s − 0.289i·4-s − 0.655·6-s + (−0.138 + 0.138i)7-s + (−0.570 − 0.570i)8-s + 0.333i·9-s + 0.624·11-s + (0.118 − 0.118i)12-s + (−0.433 − 0.433i)13-s − 0.222i·14-s + 1.20·16-s + (−0.0942 + 0.0942i)17-s + (−0.267 − 0.267i)18-s − 1.37i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.991 - 0.130i$
Analytic conductor: \(30.5533\)
Root analytic conductor: \(5.52751\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :4),\ 0.991 - 0.130i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.03745 + 0.0681262i\)
\(L(\frac12)\) \(\approx\) \(1.03745 + 0.0681262i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-33.0 - 33.0i)T \)
5 \( 1 \)
good2 \( 1 + (12.8 - 12.8i)T - 256iT^{2} \)
7 \( 1 + (333. - 333. i)T - 5.76e6iT^{2} \)
11 \( 1 - 9.14e3T + 2.14e8T^{2} \)
13 \( 1 + (1.23e4 + 1.23e4i)T + 8.15e8iT^{2} \)
17 \( 1 + (7.86e3 - 7.86e3i)T - 6.97e9iT^{2} \)
19 \( 1 + 1.78e5iT - 1.69e10T^{2} \)
23 \( 1 + (2.25e5 + 2.25e5i)T + 7.83e10iT^{2} \)
29 \( 1 + 1.02e6iT - 5.00e11T^{2} \)
31 \( 1 - 3.92e5T + 8.52e11T^{2} \)
37 \( 1 + (1.29e6 - 1.29e6i)T - 3.51e12iT^{2} \)
41 \( 1 - 3.20e6T + 7.98e12T^{2} \)
43 \( 1 + (-2.40e6 - 2.40e6i)T + 1.16e13iT^{2} \)
47 \( 1 + (-4.45e6 + 4.45e6i)T - 2.38e13iT^{2} \)
53 \( 1 + (-4.09e6 - 4.09e6i)T + 6.22e13iT^{2} \)
59 \( 1 + 1.34e7iT - 1.46e14T^{2} \)
61 \( 1 - 2.38e7T + 1.91e14T^{2} \)
67 \( 1 + (1.67e7 - 1.67e7i)T - 4.06e14iT^{2} \)
71 \( 1 - 2.63e5T + 6.45e14T^{2} \)
73 \( 1 + (3.04e7 + 3.04e7i)T + 8.06e14iT^{2} \)
79 \( 1 - 3.35e7iT - 1.51e15T^{2} \)
83 \( 1 + (4.97e7 + 4.97e7i)T + 2.25e15iT^{2} \)
89 \( 1 + 6.58e7iT - 3.93e15T^{2} \)
97 \( 1 + (-8.26e7 + 8.26e7i)T - 7.83e15iT^{2} \)
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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

−12.97742879673793126789517146558, −11.75616634610798873180400039756, −10.20924906269955446629824337028, −9.263785722689384723758934887469, −8.385119484523778872472942336126, −7.26970742861699350143816004493, −6.06968207552094593230550776949, −4.28135689753434201445990107690, −2.70256548135098186253709099952, −0.46325539058551023285431534691, 1.11735021299606935745930853245, 2.19334768467527826605448380743, 3.72141472224196461367394651175, 5.78400883324906387013919187234, 7.28846565456805462679700265427, 8.591369363516433419024710192989, 9.505275736279899601730290562795, 10.47429449431527298087315995360, 11.73944275096837177731394362163, 12.48296795884884946582457422266

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