Properties

Label 2-75-5.3-c8-0-2
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  + (−19.4 + 19.4i)2-s + (33.0 + 33.0i)3-s − 497. i·4-s − 1.28e3·6-s + (−1.85e3 + 1.85e3i)7-s + (4.69e3 + 4.69e3i)8-s + 2.18e3i·9-s + 1.55e4·11-s + (1.64e4 − 1.64e4i)12-s + (2.39e4 + 2.39e4i)13-s − 7.21e4i·14-s − 5.47e4·16-s + (−8.36e4 + 8.36e4i)17-s + (−4.24e4 − 4.24e4i)18-s + 1.33e5i·19-s + ⋯
L(s)  = 1  + (−1.21 + 1.21i)2-s + (0.408 + 0.408i)3-s − 1.94i·4-s − 0.990·6-s + (−0.773 + 0.773i)7-s + (1.14 + 1.14i)8-s + 0.333i·9-s + 1.06·11-s + (0.793 − 0.793i)12-s + (0.837 + 0.837i)13-s − 1.87i·14-s − 0.835·16-s + (−1.00 + 1.00i)17-s + (−0.404 − 0.404i)18-s + 1.02i·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: $\chi_{75} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :4),\ -0.229 + 0.973i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.319893 - 0.404202i\)
\(L(\frac12)\) \(\approx\) \(0.319893 - 0.404202i\)
\(L(5)\) 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 \( 1 + (19.4 - 19.4i)T - 256iT^{2} \)
7 \( 1 + (1.85e3 - 1.85e3i)T - 5.76e6iT^{2} \)
11 \( 1 - 1.55e4T + 2.14e8T^{2} \)
13 \( 1 + (-2.39e4 - 2.39e4i)T + 8.15e8iT^{2} \)
17 \( 1 + (8.36e4 - 8.36e4i)T - 6.97e9iT^{2} \)
19 \( 1 - 1.33e5iT - 1.69e10T^{2} \)
23 \( 1 + (2.05e5 + 2.05e5i)T + 7.83e10iT^{2} \)
29 \( 1 - 4.04e5iT - 5.00e11T^{2} \)
31 \( 1 + 6.50e5T + 8.52e11T^{2} \)
37 \( 1 + (-2.14e6 + 2.14e6i)T - 3.51e12iT^{2} \)
41 \( 1 + 8.96e5T + 7.98e12T^{2} \)
43 \( 1 + (-8.56e5 - 8.56e5i)T + 1.16e13iT^{2} \)
47 \( 1 + (1.34e6 - 1.34e6i)T - 2.38e13iT^{2} \)
53 \( 1 + (1.06e7 + 1.06e7i)T + 6.22e13iT^{2} \)
59 \( 1 + 6.21e6iT - 1.46e14T^{2} \)
61 \( 1 - 4.93e6T + 1.91e14T^{2} \)
67 \( 1 + (9.20e6 - 9.20e6i)T - 4.06e14iT^{2} \)
71 \( 1 - 5.87e6T + 6.45e14T^{2} \)
73 \( 1 + (-1.98e7 - 1.98e7i)T + 8.06e14iT^{2} \)
79 \( 1 + 2.35e7iT - 1.51e15T^{2} \)
83 \( 1 + (1.52e7 + 1.52e7i)T + 2.25e15iT^{2} \)
89 \( 1 + 5.52e6iT - 3.93e15T^{2} \)
97 \( 1 + (2.95e7 - 2.95e7i)T - 7.83e15iT^{2} \)
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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

−14.26243963047964798936146183060, −12.64124088999218977316800885500, −11.01429056324942613174708460572, −9.702187945790514894266569926776, −9.002611767394812933357922093873, −8.245246116437851560213150434137, −6.63090730287131206624618512070, −5.98044782343262387570399671683, −3.92396744978800660582694215517, −1.71893646532874796225183224645, 0.25043474931279245028976424595, 1.25476955550387122422508884802, 2.77291754233946462174757159924, 3.86831347951572827424603722632, 6.58513845145988149587668423311, 7.78006318154749737553720009586, 9.017115568919717624117591409868, 9.699695460855613375692580810596, 10.92962860003064473037930550687, 11.77874449791231227092101932242

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