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Properties

Label 2-75-5.2-c8-0-4
Degree $2$
Conductor $75$
Sign $0.973 - 0.229i$
Analytic cond. $30.5533$
Root an. cond. $5.52751$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−20.7 − 20.7i)2-s + (−33.0 + 33.0i)3-s + 604. i·4-s + 1.37e3·6-s + (500. + 500. i)7-s + (7.23e3 − 7.23e3i)8-s − 2.18e3i·9-s − 2.17e4·11-s + (−1.99e4 − 1.99e4i)12-s + (4.02e4 − 4.02e4i)13-s − 2.07e4i·14-s − 1.45e5·16-s + (−6.87e4 − 6.87e4i)17-s + (−4.53e4 + 4.53e4i)18-s + 1.99e5i·19-s + ⋯
L(s)  = 1  + (−1.29 − 1.29i)2-s + (−0.408 + 0.408i)3-s + 2.36i·4-s + 1.05·6-s + (0.208 + 0.208i)7-s + (1.76 − 1.76i)8-s − 0.333i·9-s − 1.48·11-s + (−0.964 − 0.964i)12-s + (1.40 − 1.40i)13-s − 0.540i·14-s − 2.21·16-s + (−0.823 − 0.823i)17-s + (−0.432 + 0.432i)18-s + 1.52i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.973 - 0.229i$
Analytic conductor: \(30.5533\)
Root analytic conductor: \(5.52751\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :4),\ 0.973 - 0.229i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.486744 + 0.0566735i\)
\(L(\frac12)\) \(\approx\) \(0.486744 + 0.0566735i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (33.0 - 33.0i)T \)
5 \( 1 \)
good2 \( 1 + (20.7 + 20.7i)T + 256iT^{2} \)
7 \( 1 + (-500. - 500. i)T + 5.76e6iT^{2} \)
11 \( 1 + 2.17e4T + 2.14e8T^{2} \)
13 \( 1 + (-4.02e4 + 4.02e4i)T - 8.15e8iT^{2} \)
17 \( 1 + (6.87e4 + 6.87e4i)T + 6.97e9iT^{2} \)
19 \( 1 - 1.99e5iT - 1.69e10T^{2} \)
23 \( 1 + (1.24e5 - 1.24e5i)T - 7.83e10iT^{2} \)
29 \( 1 + 6.62e5iT - 5.00e11T^{2} \)
31 \( 1 - 2.47e5T + 8.52e11T^{2} \)
37 \( 1 + (1.18e6 + 1.18e6i)T + 3.51e12iT^{2} \)
41 \( 1 - 1.13e6T + 7.98e12T^{2} \)
43 \( 1 + (4.69e5 - 4.69e5i)T - 1.16e13iT^{2} \)
47 \( 1 + (-2.35e6 - 2.35e6i)T + 2.38e13iT^{2} \)
53 \( 1 + (1.85e6 - 1.85e6i)T - 6.22e13iT^{2} \)
59 \( 1 - 1.56e7iT - 1.46e14T^{2} \)
61 \( 1 - 9.55e6T + 1.91e14T^{2} \)
67 \( 1 + (-1.27e7 - 1.27e7i)T + 4.06e14iT^{2} \)
71 \( 1 - 2.72e7T + 6.45e14T^{2} \)
73 \( 1 + (2.23e7 - 2.23e7i)T - 8.06e14iT^{2} \)
79 \( 1 - 5.20e7iT - 1.51e15T^{2} \)
83 \( 1 + (-1.12e7 + 1.12e7i)T - 2.25e15iT^{2} \)
89 \( 1 - 9.28e7iT - 3.93e15T^{2} \)
97 \( 1 + (-3.35e7 - 3.35e7i)T + 7.83e15iT^{2} \)
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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

−12.52580890043996435020704371988, −11.38774675264909161527812197568, −10.60116179562316130030039502540, −9.917088302317108696199898085122, −8.531917148077943116935024235503, −7.78117531341248534086293127836, −5.61445156025507734740735648387, −3.72016423778587031179933856690, −2.43878956090764404474130414024, −0.833474641384450957150919284914, 0.35435901700974568027700445281, 1.79555374918345209586956853535, 4.84156805167964926056576403765, 6.21783333896948462986062387978, 6.99877070685467413302964469816, 8.218701454960289144759379170732, 8.999607525118652078598216397751, 10.51869650111063577380560239812, 11.17888244104502911946705144299, 13.15008664933336572781344504894

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