Properties

Label 2-75-15.8-c7-0-29
Degree $2$
Conductor $75$
Sign $0.326 + 0.945i$
Analytic cond. $23.4288$
Root an. cond. $4.84033$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (33.0 + 33.0i)3-s − 128i·4-s + (927. − 927. i)7-s + 2.18e3i·9-s + (4.23e3 − 4.23e3i)12-s + (−6.78e3 − 6.78e3i)13-s − 1.63e4·16-s − 4.30e4i·19-s + 6.13e4·21-s + (−7.23e4 + 7.23e4i)27-s + (−1.18e5 − 1.18e5i)28-s + 3.31e5·31-s + 2.79e5·36-s + (3.88e5 − 3.88e5i)37-s − 4.48e5i·39-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)3-s − i·4-s + (1.02 − 1.02i)7-s + i·9-s + (0.707 − 0.707i)12-s + (−0.856 − 0.856i)13-s − 16-s − 1.44i·19-s + 1.44·21-s + (−0.707 + 0.707i)27-s + (−1.02 − 1.02i)28-s + 1.99·31-s + 36-s + (1.26 − 1.26i)37-s − 1.21i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.326 + 0.945i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.326 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.326 + 0.945i$
Analytic conductor: \(23.4288\)
Root analytic conductor: \(4.84033\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :7/2),\ 0.326 + 0.945i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.96049 - 1.39709i\)
\(L(\frac12)\) \(\approx\) \(1.96049 - 1.39709i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-33.0 - 33.0i)T \)
5 \( 1 \)
good2 \( 1 + 128iT^{2} \)
7 \( 1 + (-927. + 927. i)T - 8.23e5iT^{2} \)
11 \( 1 - 1.94e7T^{2} \)
13 \( 1 + (6.78e3 + 6.78e3i)T + 6.27e7iT^{2} \)
17 \( 1 + 4.10e8iT^{2} \)
19 \( 1 + 4.30e4iT - 8.93e8T^{2} \)
23 \( 1 - 3.40e9iT^{2} \)
29 \( 1 + 1.72e10T^{2} \)
31 \( 1 - 3.31e5T + 2.75e10T^{2} \)
37 \( 1 + (-3.88e5 + 3.88e5i)T - 9.49e10iT^{2} \)
41 \( 1 - 1.94e11T^{2} \)
43 \( 1 + (6.78e5 + 6.78e5i)T + 2.71e11iT^{2} \)
47 \( 1 + 5.06e11iT^{2} \)
53 \( 1 - 1.17e12iT^{2} \)
59 \( 1 + 2.48e12T^{2} \)
61 \( 1 - 1.99e6T + 3.14e12T^{2} \)
67 \( 1 + (1.97e6 - 1.97e6i)T - 6.06e12iT^{2} \)
71 \( 1 - 9.09e12T^{2} \)
73 \( 1 + (-1.55e6 - 1.55e6i)T + 1.10e13iT^{2} \)
79 \( 1 - 8.76e6iT - 1.92e13T^{2} \)
83 \( 1 - 2.71e13iT^{2} \)
89 \( 1 + 4.42e13T^{2} \)
97 \( 1 + (-2.84e6 + 2.84e6i)T - 8.07e13iT^{2} \)
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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

−13.34040470074284232635116018267, −11.35386162322091961651606869851, −10.48850827667707676077501831605, −9.711149113898568939655705125290, −8.355823144415181067062732741757, −7.15533360002861009485123078021, −5.20759614301632545135376167470, −4.37438941127491511963557531483, −2.45209306092126336171779485861, −0.76305423110524817856720216020, 1.76692303734727413390453388789, 2.88333098512087756508732556491, 4.54431658249328599439511447028, 6.44593989692216236574455233093, 7.87085441230304205755861889408, 8.372367819203288381806323777071, 9.597897021881650634476510055251, 11.81363746650167802197299951564, 11.99174528767181433576195623069, 13.26734284321356153017560167052

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