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Properties

Label 2-75-15.2-c9-0-10
Degree $2$
Conductor $75$
Sign $0.590 - 0.806i$
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  + (15.0 − 15.0i)2-s + (−124. + 65.5i)3-s + 60.0i·4-s + (−880. + 2.84e3i)6-s + (−4.27e3 − 4.27e3i)7-s + (8.59e3 + 8.59e3i)8-s + (1.10e4 − 1.62e4i)9-s − 5.87e4i·11-s + (−3.93e3 − 7.44e3i)12-s + (3.79e4 − 3.79e4i)13-s − 1.28e5·14-s + 2.27e5·16-s + (−3.16e5 + 3.16e5i)17-s + (−7.75e4 − 4.11e5i)18-s + 3.65e5i·19-s + ⋯
L(s)  = 1  + (0.664 − 0.664i)2-s + (−0.884 + 0.466i)3-s + 0.117i·4-s + (−0.277 + 0.897i)6-s + (−0.673 − 0.673i)7-s + (0.742 + 0.742i)8-s + (0.563 − 0.825i)9-s − 1.20i·11-s + (−0.0547 − 0.103i)12-s + (0.368 − 0.368i)13-s − 0.894·14-s + 0.869·16-s + (−0.919 + 0.919i)17-s + (−0.174 − 0.923i)18-s + 0.643i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.590 - 0.806i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.590 - 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.590 - 0.806i$
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: \((2,\ 75,\ (\ :9/2),\ 0.590 - 0.806i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.41801 + 0.719176i\)
\(L(\frac12)\) \(\approx\) \(1.41801 + 0.719176i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (124. - 65.5i)T \)
5 \( 1 \)
good2 \( 1 + (-15.0 + 15.0i)T - 512iT^{2} \)
7 \( 1 + (4.27e3 + 4.27e3i)T + 4.03e7iT^{2} \)
11 \( 1 + 5.87e4iT - 2.35e9T^{2} \)
13 \( 1 + (-3.79e4 + 3.79e4i)T - 1.06e10iT^{2} \)
17 \( 1 + (3.16e5 - 3.16e5i)T - 1.18e11iT^{2} \)
19 \( 1 - 3.65e5iT - 3.22e11T^{2} \)
23 \( 1 + (-1.50e6 - 1.50e6i)T + 1.80e12iT^{2} \)
29 \( 1 + 1.94e6T + 1.45e13T^{2} \)
31 \( 1 - 9.54e5T + 2.64e13T^{2} \)
37 \( 1 + (-8.41e6 - 8.41e6i)T + 1.29e14iT^{2} \)
41 \( 1 - 1.47e7iT - 3.27e14T^{2} \)
43 \( 1 + (2.54e7 - 2.54e7i)T - 5.02e14iT^{2} \)
47 \( 1 + (5.21e6 - 5.21e6i)T - 1.11e15iT^{2} \)
53 \( 1 + (8.10e6 + 8.10e6i)T + 3.29e15iT^{2} \)
59 \( 1 - 1.67e8T + 8.66e15T^{2} \)
61 \( 1 - 2.01e8T + 1.16e16T^{2} \)
67 \( 1 + (2.48e7 + 2.48e7i)T + 2.72e16iT^{2} \)
71 \( 1 + 3.74e7iT - 4.58e16T^{2} \)
73 \( 1 + (2.74e8 - 2.74e8i)T - 5.88e16iT^{2} \)
79 \( 1 - 2.73e8iT - 1.19e17T^{2} \)
83 \( 1 + (-4.89e8 - 4.89e8i)T + 1.86e17iT^{2} \)
89 \( 1 + 4.96e8T + 3.50e17T^{2} \)
97 \( 1 + (-2.56e8 - 2.56e8i)T + 7.60e17iT^{2} \)
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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

−13.01927140075085751602433646654, −11.50642997064319618731128194705, −11.00950842707428808521718117238, −9.914422169390945879749177028602, −8.300557023669369571868220335391, −6.65116446881408431990994457800, −5.43986480821180667003578670587, −4.02855498886586386871479161976, −3.25317983011705245719556298813, −1.12340126904519836081124070383, 0.47448885216060665609878282547, 2.17489039127945313036961039685, 4.45599384688108442164936438385, 5.37440787695873077145731904561, 6.62445403882397144870989838274, 7.11273500493958937149970149746, 9.148161870296029346505292842985, 10.38245125859212179140737155664, 11.61486820829738662765938615699, 12.79087500068226679249432350320

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