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Properties

Label 2-75-15.2-c7-0-15
Degree $2$
Conductor $75$
Sign $0.563 - 0.826i$
Analytic cond. $23.4288$
Root an. cond. $4.84033$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−10.4 + 10.4i)2-s + (2.10 − 46.7i)3-s − 91.1i·4-s + (467. + 511. i)6-s + (953. + 953. i)7-s + (−385. − 385. i)8-s + (−2.17e3 − 196. i)9-s − 3.82e3i·11-s + (−4.25e3 − 191. i)12-s + (−5.35e3 + 5.35e3i)13-s − 1.99e4·14-s + 1.97e4·16-s + (−2.29e3 + 2.29e3i)17-s + (2.48e4 − 2.07e4i)18-s − 4.12e4i·19-s + ⋯
L(s)  = 1  + (−0.925 + 0.925i)2-s + (0.0449 − 0.998i)3-s − 0.712i·4-s + (0.882 + 0.965i)6-s + (1.05 + 1.05i)7-s + (−0.266 − 0.266i)8-s + (−0.995 − 0.0898i)9-s − 0.865i·11-s + (−0.711 − 0.0320i)12-s + (−0.675 + 0.675i)13-s − 1.94·14-s + 1.20·16-s + (−0.113 + 0.113i)17-s + (1.00 − 0.838i)18-s − 1.38i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.563 - 0.826i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.563 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.563 - 0.826i$
Analytic conductor: \(23.4288\)
Root analytic conductor: \(4.84033\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :7/2),\ 0.563 - 0.826i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.987985 + 0.522053i\)
\(L(\frac12)\) \(\approx\) \(0.987985 + 0.522053i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.10 + 46.7i)T \)
5 \( 1 \)
good2 \( 1 + (10.4 - 10.4i)T - 128iT^{2} \)
7 \( 1 + (-953. - 953. i)T + 8.23e5iT^{2} \)
11 \( 1 + 3.82e3iT - 1.94e7T^{2} \)
13 \( 1 + (5.35e3 - 5.35e3i)T - 6.27e7iT^{2} \)
17 \( 1 + (2.29e3 - 2.29e3i)T - 4.10e8iT^{2} \)
19 \( 1 + 4.12e4iT - 8.93e8T^{2} \)
23 \( 1 + (-4.54e4 - 4.54e4i)T + 3.40e9iT^{2} \)
29 \( 1 - 1.17e5T + 1.72e10T^{2} \)
31 \( 1 + 1.02e5T + 2.75e10T^{2} \)
37 \( 1 + (-2.52e5 - 2.52e5i)T + 9.49e10iT^{2} \)
41 \( 1 - 8.02e5iT - 1.94e11T^{2} \)
43 \( 1 + (-5.15e5 + 5.15e5i)T - 2.71e11iT^{2} \)
47 \( 1 + (-1.26e5 + 1.26e5i)T - 5.06e11iT^{2} \)
53 \( 1 + (-1.43e6 - 1.43e6i)T + 1.17e12iT^{2} \)
59 \( 1 - 1.54e6T + 2.48e12T^{2} \)
61 \( 1 - 2.10e6T + 3.14e12T^{2} \)
67 \( 1 + (-1.30e5 - 1.30e5i)T + 6.06e12iT^{2} \)
71 \( 1 + 5.11e6iT - 9.09e12T^{2} \)
73 \( 1 + (-3.26e5 + 3.26e5i)T - 1.10e13iT^{2} \)
79 \( 1 - 4.71e6iT - 1.92e13T^{2} \)
83 \( 1 + (-1.64e6 - 1.64e6i)T + 2.71e13iT^{2} \)
89 \( 1 - 6.03e6T + 4.42e13T^{2} \)
97 \( 1 + (5.25e5 + 5.25e5i)T + 8.07e13iT^{2} \)
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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

−13.36736626498972154069301448798, −12.04266476243742049269079154707, −11.24037989700778228659665192894, −9.188533792676671949360807407974, −8.556107949201648656651271006487, −7.56641184673386377013890352806, −6.50909794862952988326942758063, −5.29189673717622020746425819075, −2.59554513048915725961462956287, −0.941146955707948708904550101108, 0.70646539826407515528426803333, 2.31856411351871959130118697411, 4.00923731376229086212903173282, 5.28408657183663476516836198148, 7.58239218931880736978410320987, 8.653787327022037328671021644173, 9.956772854093759417101500643410, 10.45384852534203218955053996678, 11.33680497968816609789034792059, 12.50597811843910801563528737381

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