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Properties

Label 2-75-15.2-c9-0-21
Degree $2$
Conductor $75$
Sign $0.153 - 0.988i$
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  + (−30.5 + 30.5i)2-s + (129. − 54.5i)3-s − 1.35e3i·4-s + (−2.28e3 + 5.60e3i)6-s + (2.74e3 + 2.74e3i)7-s + (2.56e4 + 2.56e4i)8-s + (1.37e4 − 1.40e4i)9-s + 4.81e3i·11-s + (−7.36e4 − 1.74e5i)12-s + (−6.67e4 + 6.67e4i)13-s − 1.67e5·14-s − 8.71e5·16-s + (−3.46e4 + 3.46e4i)17-s + (1.07e4 + 8.49e5i)18-s − 4.27e5i·19-s + ⋯
L(s)  = 1  + (−1.34 + 1.34i)2-s + (0.921 − 0.388i)3-s − 2.63i·4-s + (−0.718 + 1.76i)6-s + (0.432 + 0.432i)7-s + (2.21 + 2.21i)8-s + (0.698 − 0.715i)9-s + 0.0991i·11-s + (−1.02 − 2.43i)12-s + (−0.648 + 0.648i)13-s − 1.16·14-s − 3.32·16-s + (−0.100 + 0.100i)17-s + (0.0240 + 1.90i)18-s − 0.752i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.153 - 0.988i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.153 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.153 - 0.988i$
Analytic conductor: \(38.6276\)
Root analytic conductor: \(6.21511\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :9/2),\ 0.153 - 0.988i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.08313 + 0.927433i\)
\(L(\frac12)\) \(\approx\) \(1.08313 + 0.927433i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-129. + 54.5i)T \)
5 \( 1 \)
good2 \( 1 + (30.5 - 30.5i)T - 512iT^{2} \)
7 \( 1 + (-2.74e3 - 2.74e3i)T + 4.03e7iT^{2} \)
11 \( 1 - 4.81e3iT - 2.35e9T^{2} \)
13 \( 1 + (6.67e4 - 6.67e4i)T - 1.06e10iT^{2} \)
17 \( 1 + (3.46e4 - 3.46e4i)T - 1.18e11iT^{2} \)
19 \( 1 + 4.27e5iT - 3.22e11T^{2} \)
23 \( 1 + (-6.70e5 - 6.70e5i)T + 1.80e12iT^{2} \)
29 \( 1 + 8.83e5T + 1.45e13T^{2} \)
31 \( 1 - 7.22e6T + 2.64e13T^{2} \)
37 \( 1 + (-5.43e6 - 5.43e6i)T + 1.29e14iT^{2} \)
41 \( 1 - 2.76e7iT - 3.27e14T^{2} \)
43 \( 1 + (-1.38e7 + 1.38e7i)T - 5.02e14iT^{2} \)
47 \( 1 + (-1.97e6 + 1.97e6i)T - 1.11e15iT^{2} \)
53 \( 1 + (1.15e7 + 1.15e7i)T + 3.29e15iT^{2} \)
59 \( 1 + 2.00e7T + 8.66e15T^{2} \)
61 \( 1 - 1.28e8T + 1.16e16T^{2} \)
67 \( 1 + (-9.09e7 - 9.09e7i)T + 2.72e16iT^{2} \)
71 \( 1 + 4.40e7iT - 4.58e16T^{2} \)
73 \( 1 + (3.10e8 - 3.10e8i)T - 5.88e16iT^{2} \)
79 \( 1 + 3.03e8iT - 1.19e17T^{2} \)
83 \( 1 + (-4.46e8 - 4.46e8i)T + 1.86e17iT^{2} \)
89 \( 1 - 1.06e9T + 3.50e17T^{2} \)
97 \( 1 + (-7.40e8 - 7.40e8i)T + 7.60e17iT^{2} \)
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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

−13.38187054764889613730419407878, −11.57533846587640454715035707848, −9.993527854424436375785649352635, −9.151927786353142992487089409732, −8.310673837905455094077826488341, −7.35693527213273457700907896418, −6.43013204803021653355633451728, −4.84153698728544518203906596765, −2.25960174108194485623463679212, −0.968013279910263050214714370633, 0.75567357033840470059287035649, 2.10103654589380815519620918722, 3.16165632469876443986303676647, 4.40508463287560165557539954352, 7.43383579378105844553296905602, 8.181766351707865260994344827207, 9.208682743511812364143919684912, 10.17879795516607308054212461723, 10.85371573734087700550336207294, 12.18060021617834914589123907743

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