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Properties

Label 2-75-5.3-c8-0-1
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  + (−10.7 + 10.7i)2-s + (−33.0 − 33.0i)3-s + 24.6i·4-s + 711.·6-s + (986. − 986. i)7-s + (−3.01e3 − 3.01e3i)8-s + 2.18e3i·9-s − 7.11e3·11-s + (813. − 813. i)12-s + (2.15e4 + 2.15e4i)13-s + 2.12e4i·14-s + 5.86e4·16-s + (7.37e4 − 7.37e4i)17-s + (−2.35e4 − 2.35e4i)18-s + 1.18e5i·19-s + ⋯
L(s)  = 1  + (−0.672 + 0.672i)2-s + (−0.408 − 0.408i)3-s + 0.0961i·4-s + 0.548·6-s + (0.410 − 0.410i)7-s + (−0.736 − 0.736i)8-s + 0.333i·9-s − 0.486·11-s + (0.0392 − 0.0392i)12-s + (0.754 + 0.754i)13-s + 0.552i·14-s + 0.894·16-s + (0.882 − 0.882i)17-s + (−0.224 − 0.224i)18-s + 0.909i·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

\(L(\frac{9}{2})\) \(\approx\) \(0.0223042 - 0.339659i\)
\(L(\frac12)\) \(\approx\) \(0.0223042 - 0.339659i\)
\(L(5)\) 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 \( 1 + (10.7 - 10.7i)T - 256iT^{2} \)
7 \( 1 + (-986. + 986. i)T - 5.76e6iT^{2} \)
11 \( 1 + 7.11e3T + 2.14e8T^{2} \)
13 \( 1 + (-2.15e4 - 2.15e4i)T + 8.15e8iT^{2} \)
17 \( 1 + (-7.37e4 + 7.37e4i)T - 6.97e9iT^{2} \)
19 \( 1 - 1.18e5iT - 1.69e10T^{2} \)
23 \( 1 + (-1.15e5 - 1.15e5i)T + 7.83e10iT^{2} \)
29 \( 1 - 4.71e5iT - 5.00e11T^{2} \)
31 \( 1 + 1.10e6T + 8.52e11T^{2} \)
37 \( 1 + (4.37e5 - 4.37e5i)T - 3.51e12iT^{2} \)
41 \( 1 + 2.59e6T + 7.98e12T^{2} \)
43 \( 1 + (4.52e6 + 4.52e6i)T + 1.16e13iT^{2} \)
47 \( 1 + (1.05e6 - 1.05e6i)T - 2.38e13iT^{2} \)
53 \( 1 + (1.00e7 + 1.00e7i)T + 6.22e13iT^{2} \)
59 \( 1 + 6.33e6iT - 1.46e14T^{2} \)
61 \( 1 - 8.28e6T + 1.91e14T^{2} \)
67 \( 1 + (2.32e7 - 2.32e7i)T - 4.06e14iT^{2} \)
71 \( 1 + 3.22e7T + 6.45e14T^{2} \)
73 \( 1 + (-1.49e7 - 1.49e7i)T + 8.06e14iT^{2} \)
79 \( 1 - 7.17e7iT - 1.51e15T^{2} \)
83 \( 1 + (-2.46e7 - 2.46e7i)T + 2.25e15iT^{2} \)
89 \( 1 + 8.60e7iT - 3.93e15T^{2} \)
97 \( 1 + (7.27e7 - 7.27e7i)T - 7.83e15iT^{2} \)
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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

−13.44078792187349938330835047344, −12.29879953132091927128881093834, −11.27426849148674482251880685719, −9.934473754157277539113638624040, −8.630646542253048083120821560039, −7.61695711209732480019599235676, −6.75253023628611563461467576086, −5.35011078896583062971867639758, −3.51505618899575473283614770143, −1.37939424001100092317589935741, 0.15095410128801823167279288044, 1.53556756937533086530783466414, 3.13447205865425015754147783333, 5.06427571789579759336686181294, 6.07139095564308256548610869994, 8.061417310004810892174117970669, 9.084119306936355778068955334243, 10.28835287874379265965358148567, 10.94043277968753474700760656807, 11.92719311338006034038356921658

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