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Properties

Label 2-75-5.4-c7-0-16
Degree $2$
Conductor $75$
Sign $-0.894 + 0.447i$
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  − 6i·2-s − 27i·3-s + 92·4-s − 162·6-s + 64i·7-s − 1.32e3i·8-s − 729·9-s − 948·11-s − 2.48e3i·12-s − 5.09e3i·13-s + 384·14-s + 3.85e3·16-s − 2.83e4i·17-s + 4.37e3i·18-s + 8.62e3·19-s + ⋯
L(s)  = 1  − 0.530i·2-s − 0.577i·3-s + 0.718·4-s − 0.306·6-s + 0.0705i·7-s − 0.911i·8-s − 0.333·9-s − 0.214·11-s − 0.414i·12-s − 0.643i·13-s + 0.0374·14-s + 0.235·16-s − 1.40i·17-s + 0.176i·18-s + 0.288·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(23.4288\)
Root analytic conductor: \(4.84033\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :7/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.428700 - 1.81600i\)
\(L(\frac12)\) \(\approx\) \(0.428700 - 1.81600i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 27iT \)
5 \( 1 \)
good2 \( 1 + 6iT - 128T^{2} \)
7 \( 1 - 64iT - 8.23e5T^{2} \)
11 \( 1 + 948T + 1.94e7T^{2} \)
13 \( 1 + 5.09e3iT - 6.27e7T^{2} \)
17 \( 1 + 2.83e4iT - 4.10e8T^{2} \)
19 \( 1 - 8.62e3T + 8.93e8T^{2} \)
23 \( 1 + 1.52e4iT - 3.40e9T^{2} \)
29 \( 1 + 3.65e4T + 1.72e10T^{2} \)
31 \( 1 + 2.76e5T + 2.75e10T^{2} \)
37 \( 1 + 2.68e5iT - 9.49e10T^{2} \)
41 \( 1 + 6.29e5T + 1.94e11T^{2} \)
43 \( 1 - 6.85e5iT - 2.71e11T^{2} \)
47 \( 1 + 5.83e5iT - 5.06e11T^{2} \)
53 \( 1 + 4.28e5iT - 1.17e12T^{2} \)
59 \( 1 + 1.30e6T + 2.48e12T^{2} \)
61 \( 1 - 3.00e5T + 3.14e12T^{2} \)
67 \( 1 - 5.07e5iT - 6.06e12T^{2} \)
71 \( 1 - 5.56e6T + 9.09e12T^{2} \)
73 \( 1 - 1.36e6iT - 1.10e13T^{2} \)
79 \( 1 - 6.91e6T + 1.92e13T^{2} \)
83 \( 1 + 4.37e6iT - 2.71e13T^{2} \)
89 \( 1 - 8.52e6T + 4.42e13T^{2} \)
97 \( 1 - 8.82e6iT - 8.07e13T^{2} \)
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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

−12.49498826873465102977703193203, −11.58635786181038026526055029379, −10.64259545087159780556239378648, −9.368430594908591086876722559669, −7.76473550139272299638928441200, −6.83232463280622585182263077114, −5.40808952445470138340703521596, −3.30969596766132656198520117453, −2.10109444820072015065824644526, −0.59492147725570881658289728348, 1.89559466240585282853950335937, 3.62202687984716412447962697966, 5.28058417901085053691375727390, 6.45278599336597204987675266092, 7.70540520776019758486919745325, 8.916916967159557663492250418692, 10.34620780551102606573043851005, 11.21710719538461534120833070799, 12.35637536140373273048413582767, 13.83730537304705295522036022454

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