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Properties

Label 2-75-15.2-c9-0-25
Degree $2$
Conductor $75$
Sign $0.961 - 0.275i$
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  + (−11.0 + 11.0i)2-s + (−138. − 24.1i)3-s + 265. i·4-s + (1.80e3 − 1.26e3i)6-s + (7.39e3 + 7.39e3i)7-s + (−8.62e3 − 8.62e3i)8-s + (1.85e4 + 6.68e3i)9-s − 5.13e4i·11-s + (6.43e3 − 3.67e4i)12-s + (−1.92e4 + 1.92e4i)13-s − 1.63e5·14-s + 5.52e4·16-s + (−2.65e5 + 2.65e5i)17-s + (−2.79e5 + 1.31e5i)18-s − 8.39e5i·19-s + ⋯
L(s)  = 1  + (−0.490 + 0.490i)2-s + (−0.985 − 0.172i)3-s + 0.519i·4-s + (0.567 − 0.398i)6-s + (1.16 + 1.16i)7-s + (−0.744 − 0.744i)8-s + (0.940 + 0.339i)9-s − 1.05i·11-s + (0.0895 − 0.511i)12-s + (−0.187 + 0.187i)13-s − 1.14·14-s + 0.210·16-s + (−0.772 + 0.772i)17-s + (−0.627 + 0.294i)18-s − 1.47i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.275i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.961 - 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.961 - 0.275i$
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: \((2,\ 75,\ (\ :9/2),\ 0.961 - 0.275i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.959332 + 0.135002i\)
\(L(\frac12)\) \(\approx\) \(0.959332 + 0.135002i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (138. + 24.1i)T \)
5 \( 1 \)
good2 \( 1 + (11.0 - 11.0i)T - 512iT^{2} \)
7 \( 1 + (-7.39e3 - 7.39e3i)T + 4.03e7iT^{2} \)
11 \( 1 + 5.13e4iT - 2.35e9T^{2} \)
13 \( 1 + (1.92e4 - 1.92e4i)T - 1.06e10iT^{2} \)
17 \( 1 + (2.65e5 - 2.65e5i)T - 1.18e11iT^{2} \)
19 \( 1 + 8.39e5iT - 3.22e11T^{2} \)
23 \( 1 + (1.09e6 + 1.09e6i)T + 1.80e12iT^{2} \)
29 \( 1 - 3.71e6T + 1.45e13T^{2} \)
31 \( 1 - 4.98e6T + 2.64e13T^{2} \)
37 \( 1 + (6.94e6 + 6.94e6i)T + 1.29e14iT^{2} \)
41 \( 1 + 1.82e7iT - 3.27e14T^{2} \)
43 \( 1 + (2.43e6 - 2.43e6i)T - 5.02e14iT^{2} \)
47 \( 1 + (-1.65e7 + 1.65e7i)T - 1.11e15iT^{2} \)
53 \( 1 + (-7.36e7 - 7.36e7i)T + 3.29e15iT^{2} \)
59 \( 1 + 4.08e7T + 8.66e15T^{2} \)
61 \( 1 - 7.68e7T + 1.16e16T^{2} \)
67 \( 1 + (-9.16e7 - 9.16e7i)T + 2.72e16iT^{2} \)
71 \( 1 + 1.51e8iT - 4.58e16T^{2} \)
73 \( 1 + (-6.17e7 + 6.17e7i)T - 5.88e16iT^{2} \)
79 \( 1 + 3.41e8iT - 1.19e17T^{2} \)
83 \( 1 + (-3.79e8 - 3.79e8i)T + 1.86e17iT^{2} \)
89 \( 1 + 6.96e8T + 3.50e17T^{2} \)
97 \( 1 + (4.49e8 + 4.49e8i)T + 7.60e17iT^{2} \)
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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

−12.37476658751559820374125489529, −11.69913637395415354140585248488, −10.70401614311094519207649374506, −8.901881774640333913579864720717, −8.238616426514598226057554145845, −6.82486341345159835435828132450, −5.77570447439170583644608116167, −4.44616476179317324384695107668, −2.35281362759884430794933932010, −0.52323653648384675497588003523, 0.868553079116929379983888001792, 1.81679738452758187393218832824, 4.30403142279564521578548717337, 5.23830554433150060132552970177, 6.73762216666137310321393623675, 8.013211746835589499666996372662, 9.849493553453676386338150726510, 10.31644090674689656138707057073, 11.37905941717043494871415558956, 12.09305689625505062032165459234

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