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Properties

Label 2-75-15.14-c8-0-6
Degree $2$
Conductor $75$
Sign $0.681 - 0.732i$
Analytic cond. $30.5533$
Root an. cond. $5.52751$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 16.2·2-s + (22.8 − 77.7i)3-s + 8.51·4-s + (−371. + 1.26e3i)6-s − 569. i·7-s + 4.02e3·8-s + (−5.51e3 − 3.54e3i)9-s − 6.58e3i·11-s + (194. − 661. i)12-s + 2.82e4i·13-s + 9.26e3i·14-s − 6.76e4·16-s − 1.51e5·17-s + (8.97e4 + 5.77e4i)18-s − 4.15e4·19-s + ⋯
L(s)  = 1  − 1.01·2-s + (0.281 − 0.959i)3-s + 0.0332·4-s + (−0.286 + 0.975i)6-s − 0.237i·7-s + 0.982·8-s + (−0.841 − 0.540i)9-s − 0.449i·11-s + (0.00937 − 0.0319i)12-s + 0.987i·13-s + 0.241i·14-s − 1.03·16-s − 1.80·17-s + (0.855 + 0.549i)18-s − 0.319·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.681 - 0.732i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.681 - 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.681 - 0.732i$
Analytic conductor: \(30.5533\)
Root analytic conductor: \(5.52751\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :4),\ 0.681 - 0.732i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.469770 + 0.204582i\)
\(L(\frac12)\) \(\approx\) \(0.469770 + 0.204582i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-22.8 + 77.7i)T \)
5 \( 1 \)
good2 \( 1 + 16.2T + 256T^{2} \)
7 \( 1 + 569. iT - 5.76e6T^{2} \)
11 \( 1 + 6.58e3iT - 2.14e8T^{2} \)
13 \( 1 - 2.82e4iT - 8.15e8T^{2} \)
17 \( 1 + 1.51e5T + 6.97e9T^{2} \)
19 \( 1 + 4.15e4T + 1.69e10T^{2} \)
23 \( 1 - 1.73e5T + 7.83e10T^{2} \)
29 \( 1 + 7.83e5iT - 5.00e11T^{2} \)
31 \( 1 + 2.15e5T + 8.52e11T^{2} \)
37 \( 1 - 2.71e6iT - 3.51e12T^{2} \)
41 \( 1 - 6.76e5iT - 7.98e12T^{2} \)
43 \( 1 - 4.21e6iT - 1.16e13T^{2} \)
47 \( 1 - 8.87e6T + 2.38e13T^{2} \)
53 \( 1 - 6.53e6T + 6.22e13T^{2} \)
59 \( 1 - 1.54e7iT - 1.46e14T^{2} \)
61 \( 1 + 8.68e6T + 1.91e14T^{2} \)
67 \( 1 - 2.87e7iT - 4.06e14T^{2} \)
71 \( 1 + 3.73e7iT - 6.45e14T^{2} \)
73 \( 1 + 3.75e7iT - 8.06e14T^{2} \)
79 \( 1 + 4.22e7T + 1.51e15T^{2} \)
83 \( 1 - 7.28e7T + 2.25e15T^{2} \)
89 \( 1 - 5.99e7iT - 3.93e15T^{2} \)
97 \( 1 + 1.15e7iT - 7.83e15T^{2} \)
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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

−13.33589194542008267103754233886, −11.75218713922257943950492382308, −10.75227360968237440739957057088, −9.224741224091811073910995956355, −8.592729441416835507901111419789, −7.40799687797789078288311669471, −6.39536512374659196266691464088, −4.34595291519723211410349259998, −2.29699615513656883111568250367, −0.989995120868027835520114812320, 0.26889888583189252518412621332, 2.28755112735363505572389449110, 4.07433960535079694155380844739, 5.29375035551049666076178869585, 7.25630635982921131043173876359, 8.647602374058848069027226608621, 9.139538324965514379744180973536, 10.41084049685491111511066815890, 10.98344063781720618350773936879, 12.77320860287511095337601797208

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