↑

Properties

Label 2-75-15.2-c7-0-25
Degree $2$
Conductor $75$
Sign $0.999 + 0.0207i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (9.04 − 9.04i)2-s + (41.6 + 21.3i)3-s − 35.7i·4-s + (569. − 183. i)6-s + (−248. − 248. i)7-s + (834. + 834. i)8-s + (1.27e3 + 1.77e3i)9-s + 3.55e3i·11-s + (763. − 1.48e3i)12-s + (9.13e3 − 9.13e3i)13-s − 4.49e3·14-s + 1.96e4·16-s + (2.46e3 − 2.46e3i)17-s + (2.76e4 + 4.51e3i)18-s + 3.62e4i·19-s + ⋯
L(s)  = 1  + (0.799 − 0.799i)2-s + (0.889 + 0.456i)3-s − 0.279i·4-s + (1.07 − 0.346i)6-s + (−0.273 − 0.273i)7-s + (0.576 + 0.576i)8-s + (0.583 + 0.811i)9-s + 0.805i·11-s + (0.127 − 0.248i)12-s + (1.15 − 1.15i)13-s − 0.437·14-s + 1.20·16-s + (0.121 − 0.121i)17-s + (1.11 + 0.182i)18-s + 1.21i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0207i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0207i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.999 + 0.0207i$
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: \((2,\ 75,\ (\ :7/2),\ 0.999 + 0.0207i)\)

Particular Values

\(L(4)\) \(\approx\) \(4.30033 - 0.0445920i\)
\(L(\frac12)\) \(\approx\) \(4.30033 - 0.0445920i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-41.6 - 21.3i)T \)
5 \( 1 \)
good2 \( 1 + (-9.04 + 9.04i)T - 128iT^{2} \)
7 \( 1 + (248. + 248. i)T + 8.23e5iT^{2} \)
11 \( 1 - 3.55e3iT - 1.94e7T^{2} \)
13 \( 1 + (-9.13e3 + 9.13e3i)T - 6.27e7iT^{2} \)
17 \( 1 + (-2.46e3 + 2.46e3i)T - 4.10e8iT^{2} \)
19 \( 1 - 3.62e4iT - 8.93e8T^{2} \)
23 \( 1 + (-3.95e4 - 3.95e4i)T + 3.40e9iT^{2} \)
29 \( 1 - 6.16e4T + 1.72e10T^{2} \)
31 \( 1 + 2.27e4T + 2.75e10T^{2} \)
37 \( 1 + (1.94e5 + 1.94e5i)T + 9.49e10iT^{2} \)
41 \( 1 + 8.06e5iT - 1.94e11T^{2} \)
43 \( 1 + (6.60e5 - 6.60e5i)T - 2.71e11iT^{2} \)
47 \( 1 + (-9.11e5 + 9.11e5i)T - 5.06e11iT^{2} \)
53 \( 1 + (2.67e5 + 2.67e5i)T + 1.17e12iT^{2} \)
59 \( 1 + 2.00e6T + 2.48e12T^{2} \)
61 \( 1 + 2.16e6T + 3.14e12T^{2} \)
67 \( 1 + (-2.64e6 - 2.64e6i)T + 6.06e12iT^{2} \)
71 \( 1 + 1.38e6iT - 9.09e12T^{2} \)
73 \( 1 + (6.61e5 - 6.61e5i)T - 1.10e13iT^{2} \)
79 \( 1 + 2.77e6iT - 1.92e13T^{2} \)
83 \( 1 + (4.12e6 + 4.12e6i)T + 2.71e13iT^{2} \)
89 \( 1 + 4.51e6T + 4.42e13T^{2} \)
97 \( 1 + (3.70e6 + 3.70e6i)T + 8.07e13iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.15936251633160601586384576573, −12.26476379262619321683438217072, −10.79555520957118544348493937583, −10.05527900683668171768619590220, −8.546942222933211627557914245316, −7.46217566010495924574042243389, −5.35269451817984952413581027624, −3.94118396803767641310788650706, −3.15964509661188282311599345098, −1.66670447737764012769697323761, 1.21834767704020071402590272457, 3.11970667876427702075783975307, 4.47893615713734240077891359860, 6.18268778800503569577249850188, 6.90051010873235805195802232264, 8.376666892206366305087581084525, 9.347786689132130392571623150815, 10.99215119485538365959782346352, 12.50687202450064000045917794497, 13.62060821570923874759970408407

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