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Properties

Label 2-75-15.2-c9-0-26
Degree $2$
Conductor $75$
Sign $0.669 + 0.742i$
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  + (3.26 − 3.26i)2-s + (−38.9 − 134. i)3-s + 490. i·4-s + (−567. − 312. i)6-s + (−2.73e3 − 2.73e3i)7-s + (3.27e3 + 3.27e3i)8-s + (−1.66e4 + 1.04e4i)9-s + 6.97e3i·11-s + (6.61e4 − 1.91e4i)12-s + (−2.20e4 + 2.20e4i)13-s − 1.78e4·14-s − 2.29e5·16-s + (3.69e5 − 3.69e5i)17-s + (−2.01e4 + 8.86e4i)18-s + 4.06e5i·19-s + ⋯
L(s)  = 1  + (0.144 − 0.144i)2-s + (−0.277 − 0.960i)3-s + 0.958i·4-s + (−0.178 − 0.0985i)6-s + (−0.431 − 0.431i)7-s + (0.282 + 0.282i)8-s + (−0.846 + 0.533i)9-s + 0.143i·11-s + (0.920 − 0.265i)12-s + (−0.214 + 0.214i)13-s − 0.124·14-s − 0.876·16-s + (1.07 − 1.07i)17-s + (−0.0451 + 0.199i)18-s + 0.715i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 + 0.742i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.669 + 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.669 + 0.742i$
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: \((2,\ 75,\ (\ :9/2),\ 0.669 + 0.742i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.55069 - 0.689453i\)
\(L(\frac12)\) \(\approx\) \(1.55069 - 0.689453i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (38.9 + 134. i)T \)
5 \( 1 \)
good2 \( 1 + (-3.26 + 3.26i)T - 512iT^{2} \)
7 \( 1 + (2.73e3 + 2.73e3i)T + 4.03e7iT^{2} \)
11 \( 1 - 6.97e3iT - 2.35e9T^{2} \)
13 \( 1 + (2.20e4 - 2.20e4i)T - 1.06e10iT^{2} \)
17 \( 1 + (-3.69e5 + 3.69e5i)T - 1.18e11iT^{2} \)
19 \( 1 - 4.06e5iT - 3.22e11T^{2} \)
23 \( 1 + (-7.23e5 - 7.23e5i)T + 1.80e12iT^{2} \)
29 \( 1 - 3.65e6T + 1.45e13T^{2} \)
31 \( 1 - 2.01e6T + 2.64e13T^{2} \)
37 \( 1 + (9.47e6 + 9.47e6i)T + 1.29e14iT^{2} \)
41 \( 1 + 3.22e7iT - 3.27e14T^{2} \)
43 \( 1 + (-2.26e7 + 2.26e7i)T - 5.02e14iT^{2} \)
47 \( 1 + (-1.27e7 + 1.27e7i)T - 1.11e15iT^{2} \)
53 \( 1 + (-6.38e7 - 6.38e7i)T + 3.29e15iT^{2} \)
59 \( 1 - 1.73e8T + 8.66e15T^{2} \)
61 \( 1 + 8.75e7T + 1.16e16T^{2} \)
67 \( 1 + (-3.71e7 - 3.71e7i)T + 2.72e16iT^{2} \)
71 \( 1 + 2.72e8iT - 4.58e16T^{2} \)
73 \( 1 + (-1.26e8 + 1.26e8i)T - 5.88e16iT^{2} \)
79 \( 1 - 3.95e8iT - 1.19e17T^{2} \)
83 \( 1 + (2.69e8 + 2.69e8i)T + 1.86e17iT^{2} \)
89 \( 1 - 1.56e8T + 3.50e17T^{2} \)
97 \( 1 + (-8.99e8 - 8.99e8i)T + 7.60e17iT^{2} \)
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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

−12.31360508854367313241242292854, −11.92278907422010287962558622186, −10.49405817558250097523476599164, −8.905133022139007318802371632055, −7.59520105258966862182680188276, −6.97641714162028133339083539162, −5.35687682526999282371379864088, −3.63689613677690662537478711232, −2.35472501932899982427338279810, −0.67495559307668826443878856284, 0.879981388103270681827043515305, 2.91501538559687231475978361706, 4.51265827683498978962693234713, 5.58799330630395443025974090668, 6.51972284862997979478968821074, 8.521126895548295614606102052671, 9.719772021714665296542631291068, 10.39413209764505880722980327355, 11.49432811826810547589591730178, 12.80744286538778343303870320466

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