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Properties

Label 2-75-15.2-c9-0-24
Degree $2$
Conductor $75$
Sign $0.860 - 0.509i$
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  + (20.3 − 20.3i)2-s + (124. + 65.1i)3-s − 318. i·4-s + (3.86e3 − 1.20e3i)6-s + (−1.02e3 − 1.02e3i)7-s + (3.93e3 + 3.93e3i)8-s + (1.11e4 + 1.61e4i)9-s + 8.23e4i·11-s + (2.07e4 − 3.96e4i)12-s + (−4.61e4 + 4.61e4i)13-s − 4.17e4·14-s + 3.23e5·16-s + (−2.83e5 + 2.83e5i)17-s + (5.58e5 + 1.01e5i)18-s − 4.48e5i·19-s + ⋯
L(s)  = 1  + (0.900 − 0.900i)2-s + (0.885 + 0.464i)3-s − 0.622i·4-s + (1.21 − 0.379i)6-s + (−0.161 − 0.161i)7-s + (0.339 + 0.339i)8-s + (0.568 + 0.822i)9-s + 1.69i·11-s + (0.289 − 0.551i)12-s + (−0.448 + 0.448i)13-s − 0.290·14-s + 1.23·16-s + (−0.822 + 0.822i)17-s + (1.25 + 0.228i)18-s − 0.788i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 - 0.509i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.860 - 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.860 - 0.509i$
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: \((2,\ 75,\ (\ :9/2),\ 0.860 - 0.509i)\)

Particular Values

\(L(5)\) \(\approx\) \(4.22102 + 1.15518i\)
\(L(\frac12)\) \(\approx\) \(4.22102 + 1.15518i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-124. - 65.1i)T \)
5 \( 1 \)
good2 \( 1 + (-20.3 + 20.3i)T - 512iT^{2} \)
7 \( 1 + (1.02e3 + 1.02e3i)T + 4.03e7iT^{2} \)
11 \( 1 - 8.23e4iT - 2.35e9T^{2} \)
13 \( 1 + (4.61e4 - 4.61e4i)T - 1.06e10iT^{2} \)
17 \( 1 + (2.83e5 - 2.83e5i)T - 1.18e11iT^{2} \)
19 \( 1 + 4.48e5iT - 3.22e11T^{2} \)
23 \( 1 + (5.77e5 + 5.77e5i)T + 1.80e12iT^{2} \)
29 \( 1 - 3.17e6T + 1.45e13T^{2} \)
31 \( 1 - 4.84e6T + 2.64e13T^{2} \)
37 \( 1 + (-2.00e6 - 2.00e6i)T + 1.29e14iT^{2} \)
41 \( 1 - 2.19e7iT - 3.27e14T^{2} \)
43 \( 1 + (5.07e6 - 5.07e6i)T - 5.02e14iT^{2} \)
47 \( 1 + (-1.80e7 + 1.80e7i)T - 1.11e15iT^{2} \)
53 \( 1 + (-2.88e7 - 2.88e7i)T + 3.29e15iT^{2} \)
59 \( 1 - 1.79e8T + 8.66e15T^{2} \)
61 \( 1 + 2.05e7T + 1.16e16T^{2} \)
67 \( 1 + (1.55e8 + 1.55e8i)T + 2.72e16iT^{2} \)
71 \( 1 + 3.49e8iT - 4.58e16T^{2} \)
73 \( 1 + (5.01e7 - 5.01e7i)T - 5.88e16iT^{2} \)
79 \( 1 - 5.64e8iT - 1.19e17T^{2} \)
83 \( 1 + (2.73e8 + 2.73e8i)T + 1.86e17iT^{2} \)
89 \( 1 - 2.28e8T + 3.50e17T^{2} \)
97 \( 1 + (7.06e8 + 7.06e8i)T + 7.60e17iT^{2} \)
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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

−12.86219169464683392281720104055, −11.87925575837961417194346278147, −10.52311373036885352566363824627, −9.723346727599569163470362172113, −8.306274643882068557511422814857, −6.95490981616750089733999622866, −4.72471678805661168687058376442, −4.19968558923595275484063528506, −2.70225433720003444014816439198, −1.84678287041632684513031725424, 0.834327057285214674924442918090, 2.77110310222802934454571671856, 3.97895408442164624971804086614, 5.57147241427571302561845055626, 6.59457795658784754495311877816, 7.75155185800232727497671358109, 8.780075537041833254393508835723, 10.21066395176079991728449180359, 11.89320288962813712402919133203, 13.10779798759289451880495507569

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