Properties

Label 2-75-15.8-c9-0-40
Degree $2$
Conductor $75$
Sign $0.786 + 0.617i$
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  + (27.8 + 27.8i)2-s + (−48.2 − 131. i)3-s + 1.03e3i·4-s + (2.32e3 − 5.00e3i)6-s + (6.30e3 − 6.30e3i)7-s + (−1.45e4 + 1.45e4i)8-s + (−1.50e4 + 1.27e4i)9-s + 1.00e4i·11-s + (1.36e5 − 5.00e4i)12-s + (−1.13e5 − 1.13e5i)13-s + 3.50e5·14-s − 2.80e5·16-s + (−4.18e5 − 4.18e5i)17-s + (−7.71e5 − 6.41e4i)18-s − 7.43e4i·19-s + ⋯
L(s)  = 1  + (1.22 + 1.22i)2-s + (−0.344 − 0.938i)3-s + 2.02i·4-s + (0.731 − 1.57i)6-s + (0.991 − 0.991i)7-s + (−1.25 + 1.25i)8-s + (−0.763 + 0.646i)9-s + 0.206i·11-s + (1.89 − 0.696i)12-s + (−1.10 − 1.10i)13-s + 2.43·14-s − 1.07·16-s + (−1.21 − 1.21i)17-s + (−1.73 − 0.144i)18-s − 0.130i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.786 + 0.617i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.786 + 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.786 + 0.617i$
Analytic conductor: \(38.6276\)
Root analytic conductor: \(6.21511\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :9/2),\ 0.786 + 0.617i)\)

Particular Values

\(L(5)\) \(\approx\) \(2.76358 - 0.955862i\)
\(L(\frac12)\) \(\approx\) \(2.76358 - 0.955862i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (48.2 + 131. i)T \)
5 \( 1 \)
good2 \( 1 + (-27.8 - 27.8i)T + 512iT^{2} \)
7 \( 1 + (-6.30e3 + 6.30e3i)T - 4.03e7iT^{2} \)
11 \( 1 - 1.00e4iT - 2.35e9T^{2} \)
13 \( 1 + (1.13e5 + 1.13e5i)T + 1.06e10iT^{2} \)
17 \( 1 + (4.18e5 + 4.18e5i)T + 1.18e11iT^{2} \)
19 \( 1 + 7.43e4iT - 3.22e11T^{2} \)
23 \( 1 + (-1.15e6 + 1.15e6i)T - 1.80e12iT^{2} \)
29 \( 1 - 5.61e6T + 1.45e13T^{2} \)
31 \( 1 + 1.72e6T + 2.64e13T^{2} \)
37 \( 1 + (-5.75e6 + 5.75e6i)T - 1.29e14iT^{2} \)
41 \( 1 + 1.54e7iT - 3.27e14T^{2} \)
43 \( 1 + (2.43e7 + 2.43e7i)T + 5.02e14iT^{2} \)
47 \( 1 + (-8.82e6 - 8.82e6i)T + 1.11e15iT^{2} \)
53 \( 1 + (-4.01e7 + 4.01e7i)T - 3.29e15iT^{2} \)
59 \( 1 + 1.04e8T + 8.66e15T^{2} \)
61 \( 1 - 1.51e7T + 1.16e16T^{2} \)
67 \( 1 + (5.07e7 - 5.07e7i)T - 2.72e16iT^{2} \)
71 \( 1 - 4.26e7iT - 4.58e16T^{2} \)
73 \( 1 + (-3.98e7 - 3.98e7i)T + 5.88e16iT^{2} \)
79 \( 1 - 1.98e8iT - 1.19e17T^{2} \)
83 \( 1 + (1.82e7 - 1.82e7i)T - 1.86e17iT^{2} \)
89 \( 1 + 7.56e8T + 3.50e17T^{2} \)
97 \( 1 + (8.05e8 - 8.05e8i)T - 7.60e17iT^{2} \)
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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

−12.90014538313590266772780887703, −11.94339782207126203765316909479, −10.69808599670021266774760290397, −8.368794124796850653279191210780, −7.33927254407786216111633480099, −6.84272871712439638633714577473, −5.25807122056850958082270608498, −4.58248016494900505826951210991, −2.62861816180396362912988688916, −0.56303230404389863026765954605, 1.66469257973600752730153532742, 2.85409522794290105550226289624, 4.38511802221405672294036168132, 4.96040054902942787911004211191, 6.18635062355392256392012948448, 8.684623119506683713137968921587, 9.841166206575669212428055201006, 11.05883568404250154461274311233, 11.60203084029701122404631645942, 12.45192735629683280311559610553

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