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Properties

Label 2-75-5.2-c8-0-9
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  + (−3.10 − 3.10i)2-s + (−33.0 + 33.0i)3-s − 236. i·4-s + 205.·6-s + (974. + 974. i)7-s + (−1.53e3 + 1.53e3i)8-s − 2.18e3i·9-s − 4.99e3·11-s + (7.82e3 + 7.82e3i)12-s + (−1.49e4 + 1.49e4i)13-s − 6.05e3i·14-s − 5.10e4·16-s + (1.17e4 + 1.17e4i)17-s + (−6.79e3 + 6.79e3i)18-s − 5.25e4i·19-s + ⋯
L(s)  = 1  + (−0.194 − 0.194i)2-s + (−0.408 + 0.408i)3-s − 0.924i·4-s + 0.158·6-s + (0.405 + 0.405i)7-s + (−0.373 + 0.373i)8-s − 0.333i·9-s − 0.341·11-s + (0.377 + 0.377i)12-s + (−0.523 + 0.523i)13-s − 0.157i·14-s − 0.779·16-s + (0.141 + 0.141i)17-s + (−0.0647 + 0.0647i)18-s − 0.403i·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

\(L(\frac{9}{2})\) \(\approx\) \(1.33316 + 0.155225i\)
\(L(\frac12)\) \(\approx\) \(1.33316 + 0.155225i\)
\(L(5)\) 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 \( 1 + (3.10 + 3.10i)T + 256iT^{2} \)
7 \( 1 + (-974. - 974. i)T + 5.76e6iT^{2} \)
11 \( 1 + 4.99e3T + 2.14e8T^{2} \)
13 \( 1 + (1.49e4 - 1.49e4i)T - 8.15e8iT^{2} \)
17 \( 1 + (-1.17e4 - 1.17e4i)T + 6.97e9iT^{2} \)
19 \( 1 + 5.25e4iT - 1.69e10T^{2} \)
23 \( 1 + (-2.45e5 + 2.45e5i)T - 7.83e10iT^{2} \)
29 \( 1 - 1.02e6iT - 5.00e11T^{2} \)
31 \( 1 + 1.65e5T + 8.52e11T^{2} \)
37 \( 1 + (-2.47e6 - 2.47e6i)T + 3.51e12iT^{2} \)
41 \( 1 - 3.10e6T + 7.98e12T^{2} \)
43 \( 1 + (-1.63e6 + 1.63e6i)T - 1.16e13iT^{2} \)
47 \( 1 + (-5.87e6 - 5.87e6i)T + 2.38e13iT^{2} \)
53 \( 1 + (-8.41e6 + 8.41e6i)T - 6.22e13iT^{2} \)
59 \( 1 + 9.42e6iT - 1.46e14T^{2} \)
61 \( 1 - 3.74e6T + 1.91e14T^{2} \)
67 \( 1 + (-1.23e7 - 1.23e7i)T + 4.06e14iT^{2} \)
71 \( 1 - 1.38e7T + 6.45e14T^{2} \)
73 \( 1 + (2.14e7 - 2.14e7i)T - 8.06e14iT^{2} \)
79 \( 1 - 4.04e7iT - 1.51e15T^{2} \)
83 \( 1 + (1.85e7 - 1.85e7i)T - 2.25e15iT^{2} \)
89 \( 1 + 5.33e7iT - 3.93e15T^{2} \)
97 \( 1 + (8.55e7 + 8.55e7i)T + 7.83e15iT^{2} \)
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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

−12.76636561492496873670410938384, −11.50635020359376485834009146158, −10.73318543476331223708394859340, −9.667128197692971102289461450842, −8.667541505940664723643041124673, −6.87038295056554448439718142257, −5.54647006000888840541337271452, −4.61312136324821150124824854816, −2.48604894736160365742898166555, −0.913155551037018025213283511916, 0.64049231231612469305516137913, 2.56358985999048050156391030811, 4.18773605159515717098340125131, 5.74415153335701771815650084800, 7.34351232586092343024307067697, 7.84752487354757139363387907967, 9.321618988759440649484752044837, 10.78091235086279068419428010140, 11.83982922820622720024330200839, 12.78579047248581285320828635280

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