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Properties

Label 2-75-15.2-c11-0-17
Degree $2$
Conductor $75$
Sign $0.0441 - 0.999i$
Analytic cond. $57.6257$
Root an. cond. $7.59116$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (21.4 − 21.4i)2-s + (−367. + 205. i)3-s + 1.12e3i·4-s + (−3.47e3 + 1.22e4i)6-s + (−4.53e4 − 4.53e4i)7-s + (6.80e4 + 6.80e4i)8-s + (9.29e4 − 1.50e5i)9-s + 6.45e5i·11-s + (−2.31e5 − 4.14e5i)12-s + (9.46e5 − 9.46e5i)13-s − 1.94e6·14-s + 6.06e5·16-s + (7.24e6 − 7.24e6i)17-s + (−1.24e6 − 5.22e6i)18-s + 4.63e6i·19-s + ⋯
L(s)  = 1  + (0.473 − 0.473i)2-s + (−0.873 + 0.487i)3-s + 0.551i·4-s + (−0.182 + 0.644i)6-s + (−1.02 − 1.02i)7-s + (0.734 + 0.734i)8-s + (0.524 − 0.851i)9-s + 1.20i·11-s + (−0.268 − 0.481i)12-s + (0.707 − 0.707i)13-s − 0.967·14-s + 0.144·16-s + (1.23 − 1.23i)17-s + (−0.154 − 0.651i)18-s + 0.429i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0441 - 0.999i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.0441 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.0441 - 0.999i$
Analytic conductor: \(57.6257\)
Root analytic conductor: \(7.59116\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :11/2),\ 0.0441 - 0.999i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.977328 + 0.935047i\)
\(L(\frac12)\) \(\approx\) \(0.977328 + 0.935047i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (367. - 205. i)T \)
5 \( 1 \)
good2 \( 1 + (-21.4 + 21.4i)T - 2.04e3iT^{2} \)
7 \( 1 + (4.53e4 + 4.53e4i)T + 1.97e9iT^{2} \)
11 \( 1 - 6.45e5iT - 2.85e11T^{2} \)
13 \( 1 + (-9.46e5 + 9.46e5i)T - 1.79e12iT^{2} \)
17 \( 1 + (-7.24e6 + 7.24e6i)T - 3.42e13iT^{2} \)
19 \( 1 - 4.63e6iT - 1.16e14T^{2} \)
23 \( 1 + (7.22e6 + 7.22e6i)T + 9.52e14iT^{2} \)
29 \( 1 - 7.09e7T + 1.22e16T^{2} \)
31 \( 1 + 1.10e8T + 2.54e16T^{2} \)
37 \( 1 + (1.47e7 + 1.47e7i)T + 1.77e17iT^{2} \)
41 \( 1 - 5.09e8iT - 5.50e17T^{2} \)
43 \( 1 + (9.36e8 - 9.36e8i)T - 9.29e17iT^{2} \)
47 \( 1 + (1.38e9 - 1.38e9i)T - 2.47e18iT^{2} \)
53 \( 1 + (-7.45e8 - 7.45e8i)T + 9.26e18iT^{2} \)
59 \( 1 - 3.69e9T + 3.01e19T^{2} \)
61 \( 1 + 4.33e8T + 4.35e19T^{2} \)
67 \( 1 + (-1.02e10 - 1.02e10i)T + 1.22e20iT^{2} \)
71 \( 1 - 1.01e10iT - 2.31e20T^{2} \)
73 \( 1 + (-8.86e9 + 8.86e9i)T - 3.13e20iT^{2} \)
79 \( 1 - 1.93e10iT - 7.47e20T^{2} \)
83 \( 1 + (1.92e10 + 1.92e10i)T + 1.28e21iT^{2} \)
89 \( 1 - 3.40e9T + 2.77e21T^{2} \)
97 \( 1 + (-3.61e10 - 3.61e10i)T + 7.15e21iT^{2} \)
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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

−12.51713375554520178932856708928, −11.55983906230381562368160919934, −10.37239007949762575626515792614, −9.693987342728610333780416310806, −7.72855771927271876029018704944, −6.67243776095628889700266971249, −5.13032497114777232048710475567, −4.00835485282217705086756188141, −3.10482788588494390350925574749, −1.05021479564477896596030879801, 0.39299807338966286674206304495, 1.69186629856477826148078491727, 3.61679130681249739622350163999, 5.39973260711426408435872589908, 6.00887846303170344806535751085, 6.77501751024219555261596143600, 8.489391762144086117078773423906, 9.911466267802427346177293963678, 10.99556092736609492472570209640, 12.14472272867259831467977839747

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