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Properties

Label 2-75-15.2-c11-0-20
Degree $2$
Conductor $75$
Sign $0.808 - 0.588i$
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  + (−26.1 + 26.1i)2-s + (−389. − 159. i)3-s + 681. i·4-s + (1.43e4 − 6.02e3i)6-s + (−5.51e3 − 5.51e3i)7-s + (−7.13e4 − 7.13e4i)8-s + (1.26e5 + 1.24e5i)9-s + 2.13e5i·11-s + (1.08e5 − 2.65e5i)12-s + (−1.40e6 + 1.40e6i)13-s + 2.88e5·14-s + 2.33e6·16-s + (2.24e6 − 2.24e6i)17-s + (−6.54e6 + 6.07e4i)18-s − 1.02e7i·19-s + ⋯
L(s)  = 1  + (−0.577 + 0.577i)2-s + (−0.925 − 0.378i)3-s + 0.332i·4-s + (0.753 − 0.316i)6-s + (−0.123 − 0.123i)7-s + (−0.769 − 0.769i)8-s + (0.713 + 0.700i)9-s + 0.398i·11-s + (0.125 − 0.308i)12-s + (−1.05 + 1.05i)13-s + 0.143·14-s + 0.556·16-s + (0.383 − 0.383i)17-s + (−0.816 + 0.00757i)18-s − 0.948i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.808 - 0.588i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.808 - 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.808 - 0.588i$
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: \((2,\ 75,\ (\ :11/2),\ 0.808 - 0.588i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.501658 + 0.163231i\)
\(L(\frac12)\) \(\approx\) \(0.501658 + 0.163231i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (389. + 159. i)T \)
5 \( 1 \)
good2 \( 1 + (26.1 - 26.1i)T - 2.04e3iT^{2} \)
7 \( 1 + (5.51e3 + 5.51e3i)T + 1.97e9iT^{2} \)
11 \( 1 - 2.13e5iT - 2.85e11T^{2} \)
13 \( 1 + (1.40e6 - 1.40e6i)T - 1.79e12iT^{2} \)
17 \( 1 + (-2.24e6 + 2.24e6i)T - 3.42e13iT^{2} \)
19 \( 1 + 1.02e7iT - 1.16e14T^{2} \)
23 \( 1 + (1.46e7 + 1.46e7i)T + 9.52e14iT^{2} \)
29 \( 1 + 1.52e8T + 1.22e16T^{2} \)
31 \( 1 + 2.64e8T + 2.54e16T^{2} \)
37 \( 1 + (2.07e8 + 2.07e8i)T + 1.77e17iT^{2} \)
41 \( 1 - 1.29e8iT - 5.50e17T^{2} \)
43 \( 1 + (-2.00e8 + 2.00e8i)T - 9.29e17iT^{2} \)
47 \( 1 + (-9.40e8 + 9.40e8i)T - 2.47e18iT^{2} \)
53 \( 1 + (-3.25e9 - 3.25e9i)T + 9.26e18iT^{2} \)
59 \( 1 - 4.71e9T + 3.01e19T^{2} \)
61 \( 1 + 4.47e9T + 4.35e19T^{2} \)
67 \( 1 + (-1.40e10 - 1.40e10i)T + 1.22e20iT^{2} \)
71 \( 1 - 9.83e9iT - 2.31e20T^{2} \)
73 \( 1 + (2.37e10 - 2.37e10i)T - 3.13e20iT^{2} \)
79 \( 1 - 6.03e9iT - 7.47e20T^{2} \)
83 \( 1 + (3.18e10 + 3.18e10i)T + 1.28e21iT^{2} \)
89 \( 1 - 9.38e10T + 2.77e21T^{2} \)
97 \( 1 + (-1.56e10 - 1.56e10i)T + 7.15e21iT^{2} \)
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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

−12.28422477588805674551102484014, −11.43126471763849086823195800575, −9.981543910469200562020233752934, −8.954527718316521929386192285302, −7.28414420144978619607064696911, −7.05797850961718830284159532149, −5.53243999001319121077655649550, −4.11613916554502023670286390362, −2.16499907438422190512787584375, −0.40765883168801150566437738775, 0.47647719008657328065756593452, 1.79688847577740858153654063272, 3.49880883151146056822682484491, 5.28653358350652111349669873762, 5.93828859828672130964621519288, 7.65484668144710614004794645307, 9.234917229134729726713403861474, 10.11660184110322358121412596174, 10.83923411003687953594049698839, 11.89886147445442291480473236045

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