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Properties

Label 2-75-5.2-c8-0-8
Degree $2$
Conductor $75$
Sign $0.229 + 0.973i$
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  + (−17.1 − 17.1i)2-s + (−33.0 + 33.0i)3-s + 333. i·4-s + 1.13e3·6-s + (−1.26e3 − 1.26e3i)7-s + (1.32e3 − 1.32e3i)8-s − 2.18e3i·9-s − 1.92e4·11-s + (−1.10e4 − 1.10e4i)12-s + (−2.99e4 + 2.99e4i)13-s + 4.35e4i·14-s + 3.98e4·16-s + (4.26e4 + 4.26e4i)17-s + (−3.75e4 + 3.75e4i)18-s − 8.64e4i·19-s + ⋯
L(s)  = 1  + (−1.07 − 1.07i)2-s + (−0.408 + 0.408i)3-s + 1.30i·4-s + 0.875·6-s + (−0.528 − 0.528i)7-s + (0.323 − 0.323i)8-s − 0.333i·9-s − 1.31·11-s + (−0.531 − 0.531i)12-s + (−1.04 + 1.04i)13-s + 1.13i·14-s + 0.607·16-s + (0.510 + 0.510i)17-s + (−0.357 + 0.357i)18-s − 0.663i·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.229 + 0.973i$
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: \((2,\ 75,\ (\ :4),\ 0.229 + 0.973i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.305648 - 0.241896i\)
\(L(\frac12)\) \(\approx\) \(0.305648 - 0.241896i\)
\(L(5)\) 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 \( 1 + (17.1 + 17.1i)T + 256iT^{2} \)
7 \( 1 + (1.26e3 + 1.26e3i)T + 5.76e6iT^{2} \)
11 \( 1 + 1.92e4T + 2.14e8T^{2} \)
13 \( 1 + (2.99e4 - 2.99e4i)T - 8.15e8iT^{2} \)
17 \( 1 + (-4.26e4 - 4.26e4i)T + 6.97e9iT^{2} \)
19 \( 1 + 8.64e4iT - 1.69e10T^{2} \)
23 \( 1 + (2.13e5 - 2.13e5i)T - 7.83e10iT^{2} \)
29 \( 1 - 6.25e5iT - 5.00e11T^{2} \)
31 \( 1 - 8.44e4T + 8.52e11T^{2} \)
37 \( 1 + (-1.16e6 - 1.16e6i)T + 3.51e12iT^{2} \)
41 \( 1 + 4.76e6T + 7.98e12T^{2} \)
43 \( 1 + (-2.56e6 + 2.56e6i)T - 1.16e13iT^{2} \)
47 \( 1 + (5.81e6 + 5.81e6i)T + 2.38e13iT^{2} \)
53 \( 1 + (-7.04e6 + 7.04e6i)T - 6.22e13iT^{2} \)
59 \( 1 + 6.85e6iT - 1.46e14T^{2} \)
61 \( 1 - 2.17e7T + 1.91e14T^{2} \)
67 \( 1 + (-3.01e6 - 3.01e6i)T + 4.06e14iT^{2} \)
71 \( 1 - 1.55e7T + 6.45e14T^{2} \)
73 \( 1 + (4.88e6 - 4.88e6i)T - 8.06e14iT^{2} \)
79 \( 1 - 2.98e6iT - 1.51e15T^{2} \)
83 \( 1 + (-3.64e7 + 3.64e7i)T - 2.25e15iT^{2} \)
89 \( 1 - 4.15e7iT - 3.93e15T^{2} \)
97 \( 1 + (-3.41e7 - 3.41e7i)T + 7.83e15iT^{2} \)
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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

−12.21422825871990926344043291754, −11.30490657069491934655554584201, −10.16549276780093412475796717612, −9.791845945749481621276528909786, −8.397994154526285698107794498340, −7.04421608107591939655883279423, −5.19388407723210140855131300619, −3.49462135342296495606680011327, −2.06914530718181351183536563880, −0.37711839188489999678105939770, 0.52691082583818376346028147801, 2.66059387773911979638305386722, 5.29780713726643299163472162414, 6.18883326247018302179416040885, 7.55974527706487328172852994850, 8.137825342973700931561860966270, 9.681175915783643550752703755277, 10.38943823325759326211311688231, 12.15287844490150095644646109527, 12.97480501608736365715422612941

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