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Properties

Label 2-75-5.3-c8-0-18
Degree $2$
Conductor $75$
Sign $0.326 + 0.945i$
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  + (−17.7 + 17.7i)2-s + (−33.0 − 33.0i)3-s − 373. i·4-s + 1.17e3·6-s + (3.27e3 − 3.27e3i)7-s + (2.09e3 + 2.09e3i)8-s + 2.18e3i·9-s + 1.78e4·11-s + (−1.23e4 + 1.23e4i)12-s + (−2.10e4 − 2.10e4i)13-s + 1.16e5i·14-s + 2.14e4·16-s + (−1.10e4 + 1.10e4i)17-s + (−3.88e4 − 3.88e4i)18-s − 4.19e3i·19-s + ⋯
L(s)  = 1  + (−1.10 + 1.10i)2-s + (−0.408 − 0.408i)3-s − 1.46i·4-s + 0.905·6-s + (1.36 − 1.36i)7-s + (0.510 + 0.510i)8-s + 0.333i·9-s + 1.21·11-s + (−0.596 + 0.596i)12-s + (−0.736 − 0.736i)13-s + 3.02i·14-s + 0.327·16-s + (−0.132 + 0.132i)17-s + (−0.369 − 0.369i)18-s − 0.0321i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.326 + 0.945i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.326 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.326 + 0.945i$
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: \((2,\ 75,\ (\ :4),\ 0.326 + 0.945i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.667474 - 0.475656i\)
\(L(\frac12)\) \(\approx\) \(0.667474 - 0.475656i\)
\(L(5)\) 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 \( 1 + (17.7 - 17.7i)T - 256iT^{2} \)
7 \( 1 + (-3.27e3 + 3.27e3i)T - 5.76e6iT^{2} \)
11 \( 1 - 1.78e4T + 2.14e8T^{2} \)
13 \( 1 + (2.10e4 + 2.10e4i)T + 8.15e8iT^{2} \)
17 \( 1 + (1.10e4 - 1.10e4i)T - 6.97e9iT^{2} \)
19 \( 1 + 4.19e3iT - 1.69e10T^{2} \)
23 \( 1 + (1.94e5 + 1.94e5i)T + 7.83e10iT^{2} \)
29 \( 1 - 7.28e5iT - 5.00e11T^{2} \)
31 \( 1 + 1.11e6T + 8.52e11T^{2} \)
37 \( 1 + (-2.27e6 + 2.27e6i)T - 3.51e12iT^{2} \)
41 \( 1 - 5.27e6T + 7.98e12T^{2} \)
43 \( 1 + (-1.31e6 - 1.31e6i)T + 1.16e13iT^{2} \)
47 \( 1 + (-4.96e5 + 4.96e5i)T - 2.38e13iT^{2} \)
53 \( 1 + (-1.05e6 - 1.05e6i)T + 6.22e13iT^{2} \)
59 \( 1 + 2.25e6iT - 1.46e14T^{2} \)
61 \( 1 + 1.64e7T + 1.91e14T^{2} \)
67 \( 1 + (1.76e7 - 1.76e7i)T - 4.06e14iT^{2} \)
71 \( 1 + 7.32e6T + 6.45e14T^{2} \)
73 \( 1 + (1.61e7 + 1.61e7i)T + 8.06e14iT^{2} \)
79 \( 1 - 1.60e7iT - 1.51e15T^{2} \)
83 \( 1 + (6.51e7 + 6.51e7i)T + 2.25e15iT^{2} \)
89 \( 1 + 1.05e8iT - 3.93e15T^{2} \)
97 \( 1 + (-2.43e7 + 2.43e7i)T - 7.83e15iT^{2} \)
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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

−12.64773460190908360772336963182, −11.20734270912060744997421566226, −10.31560260799203456295818528652, −8.950654260797172787214832188883, −7.69135322903075959976300989555, −7.23241787871718538190744283014, −5.88307226222941811152176493010, −4.35001772834428644144525356459, −1.44002864298399061713364544128, −0.44320527509268204140051672801, 1.37107366248809536657453261470, 2.39359205418089616381020859019, 4.30443245089098977816805317635, 5.82358454214560596887031762031, 7.82231424967025417436067279142, 9.047400641023098208153197766994, 9.556232367311740999298568007031, 11.09624384748139040880335810606, 11.69007697570738796321367218228, 12.25046904476772989167328519285

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