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Properties

Label 2-75-5.3-c8-0-22
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: $\chi_{75} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :4),\ -0.973 - 0.229i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.334532 + 2.87315i\)
\(L(\frac12)\) \(\approx\) \(0.334532 + 2.87315i\)
\(L(5)\) 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

−12.60114586656226347277231703813, −11.23459635896821692367412945150, −10.31433052953433688220697849289, −9.490723808185180205966100005971, −7.56002615650674911446687053026, −5.39051922284285493808314475897, −4.86497976522912141045864935698, −3.00795172328509703705906214110, −2.62661248531638457377039495823, −0.51019784862200433906897018539, 2.45047188648397308908985977396, 3.90057347727414532035870296712, 5.15715874955092946739969018610, 6.37896558509053252212082045143, 7.49622158094397183533089071672, 8.196673385680959328636981417643, 9.977001837368398921044988074584, 11.99846325568659183373421735315, 12.72413981688951152787332145073, 13.66576899793599037048880992420

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