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Properties

Label 2-75-15.2-c3-0-14
Degree $2$
Conductor $75$
Sign $-0.945 + 0.326i$
Analytic cond. $4.42514$
Root an. cond. $2.10360$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.39 − 3.39i)2-s + (−5.15 − 0.624i)3-s − 15.1i·4-s + (−19.6 + 15.4i)6-s + (−19.7 − 19.7i)7-s + (−24.1 − 24.1i)8-s + (26.2 + 6.44i)9-s + 9.19i·11-s + (−9.43 + 77.9i)12-s + (22.4 − 22.4i)13-s − 134.·14-s − 43.3·16-s + (50.6 − 50.6i)17-s + (111. − 67.1i)18-s − 16.5i·19-s + ⋯
L(s)  = 1  + (1.20 − 1.20i)2-s + (−0.992 − 0.120i)3-s − 1.88i·4-s + (−1.33 + 1.04i)6-s + (−1.06 − 1.06i)7-s + (−1.06 − 1.06i)8-s + (0.971 + 0.238i)9-s + 0.252i·11-s + (−0.227 + 1.87i)12-s + (0.478 − 0.478i)13-s − 2.55·14-s − 0.676·16-s + (0.722 − 0.722i)17-s + (1.45 − 0.879i)18-s − 0.200i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 + 0.326i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.945 + 0.326i$
Analytic conductor: \(4.42514\)
Root analytic conductor: \(2.10360\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :3/2),\ -0.945 + 0.326i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.288614 - 1.72090i\)
\(L(\frac12)\) \(\approx\) \(0.288614 - 1.72090i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (5.15 + 0.624i)T \)
5 \( 1 \)
good2 \( 1 + (-3.39 + 3.39i)T - 8iT^{2} \)
7 \( 1 + (19.7 + 19.7i)T + 343iT^{2} \)
11 \( 1 - 9.19iT - 1.33e3T^{2} \)
13 \( 1 + (-22.4 + 22.4i)T - 2.19e3iT^{2} \)
17 \( 1 + (-50.6 + 50.6i)T - 4.91e3iT^{2} \)
19 \( 1 + 16.5iT - 6.85e3T^{2} \)
23 \( 1 + (-48.2 - 48.2i)T + 1.21e4iT^{2} \)
29 \( 1 - 203.T + 2.43e4T^{2} \)
31 \( 1 + 27.4T + 2.97e4T^{2} \)
37 \( 1 + (130. + 130. i)T + 5.06e4iT^{2} \)
41 \( 1 + 9.19iT - 6.89e4T^{2} \)
43 \( 1 + (63.3 - 63.3i)T - 7.95e4iT^{2} \)
47 \( 1 + (383. - 383. i)T - 1.03e5iT^{2} \)
53 \( 1 + (-441. - 441. i)T + 1.48e5iT^{2} \)
59 \( 1 - 314.T + 2.05e5T^{2} \)
61 \( 1 + 431.T + 2.26e5T^{2} \)
67 \( 1 + (-649. - 649. i)T + 3.00e5iT^{2} \)
71 \( 1 + 722. iT - 3.57e5T^{2} \)
73 \( 1 + (662. - 662. i)T - 3.89e5iT^{2} \)
79 \( 1 + 206. iT - 4.93e5T^{2} \)
83 \( 1 + (544. + 544. i)T + 5.71e5iT^{2} \)
89 \( 1 - 563.T + 7.04e5T^{2} \)
97 \( 1 + (-66.0 - 66.0i)T + 9.12e5iT^{2} \)
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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

−13.21183015670021722692927013400, −12.54214451363558893713952420483, −11.51236831014425583568796597465, −10.51247346892184258610397819462, −9.848030224827326367614698880001, −7.16342939933659941103240126475, −5.88070786053396415433309354042, −4.58554979227222948386084569142, −3.27596742995708976857112768916, −0.930323177065386172254461674217, 3.56005663386331909213289826760, 5.11776401265818888806440272110, 6.11185691585635080120519115794, 6.76647100727266038727672519237, 8.508447523656792253103465966840, 10.11579813678626037977055123493, 11.81130771760501889559357202935, 12.54250098591849595670039792733, 13.38995544600361351035120351887, 14.74761977133898377178248665282

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