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Properties

Label 2-75-15.2-c3-0-15
Degree $2$
Conductor $75$
Sign $-0.714 + 0.699i$
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 + (−0.624 − 5.15i)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 + (−77.9 + 9.43i)12-s + (−22.4 + 22.4i)13-s + 134.·14-s − 43.3·16-s + (50.6 − 50.6i)17-s + (−67.1 + 111. i)18-s − 16.5i·19-s + ⋯
L(s)  = 1  + (1.20 − 1.20i)2-s + (−0.120 − 0.992i)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 + (−1.87 + 0.227i)12-s + (−0.478 + 0.478i)13-s + 2.55·14-s − 0.676·16-s + (0.722 − 0.722i)17-s + (−0.879 + 1.45i)18-s − 0.200i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.714 + 0.699i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.714 + 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.714 + 0.699i$
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: \((2,\ 75,\ (\ :3/2),\ -0.714 + 0.699i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.00297 - 2.45808i\)
\(L(\frac12)\) \(\approx\) \(1.00297 - 2.45808i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.624 + 5.15i)T \)
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

−13.45134728475049492128055524732, −12.37282677025495094694947496317, −11.71103081809632663740421327641, −11.05401617233440073616044840382, −9.238748592323060773045441197349, −7.68400134438767800992911775564, −5.85220201348377979026166773595, −4.91328664437077994917584110783, −2.83366169353053348271232098051, −1.59527154563366239663434728659, 3.69609945633593234426254259562, 4.69541200206665638893437539120, 5.67935723028236458463122939318, 7.28941900761324452588403863940, 8.275106769127800018240390815539, 10.09565099927701054298710316683, 11.23011328994391148092987404188, 12.60240722648898611866401051331, 13.81318721370557735616956517025, 14.80846692841384243062480519334

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