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Properties

Label 2-75-15.2-c9-0-38
Degree $2$
Conductor $75$
Sign $-0.931 + 0.363i$
Analytic cond. $38.6276$
Root an. cond. $6.21511$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (10.0 − 10.0i)2-s + (−112. − 84.4i)3-s + 311. i·4-s + (−1.96e3 + 277. i)6-s + (−2.70e3 − 2.70e3i)7-s + (8.24e3 + 8.24e3i)8-s + (5.43e3 + 1.89e4i)9-s + 2.51e4i·11-s + (2.62e4 − 3.48e4i)12-s + (1.31e5 − 1.31e5i)13-s − 5.42e4·14-s + 6.08e3·16-s + (−2.74e5 + 2.74e5i)17-s + (2.44e5 + 1.35e5i)18-s − 5.04e5i·19-s + ⋯
L(s)  = 1  + (0.442 − 0.442i)2-s + (−0.798 − 0.601i)3-s + 0.607i·4-s + (−0.620 + 0.0872i)6-s + (−0.426 − 0.426i)7-s + (0.712 + 0.712i)8-s + (0.275 + 0.961i)9-s + 0.518i·11-s + (0.365 − 0.485i)12-s + (1.28 − 1.28i)13-s − 0.377·14-s + 0.0231·16-s + (−0.796 + 0.796i)17-s + (0.547 + 0.303i)18-s − 0.887i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.931 + 0.363i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.931 + 0.363i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.931 + 0.363i$
Analytic conductor: \(38.6276\)
Root analytic conductor: \(6.21511\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :9/2),\ -0.931 + 0.363i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.157252 - 0.836605i\)
\(L(\frac12)\) \(\approx\) \(0.157252 - 0.836605i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (112. + 84.4i)T \)
5 \( 1 \)
good2 \( 1 + (-10.0 + 10.0i)T - 512iT^{2} \)
7 \( 1 + (2.70e3 + 2.70e3i)T + 4.03e7iT^{2} \)
11 \( 1 - 2.51e4iT - 2.35e9T^{2} \)
13 \( 1 + (-1.31e5 + 1.31e5i)T - 1.06e10iT^{2} \)
17 \( 1 + (2.74e5 - 2.74e5i)T - 1.18e11iT^{2} \)
19 \( 1 + 5.04e5iT - 3.22e11T^{2} \)
23 \( 1 + (3.57e5 + 3.57e5i)T + 1.80e12iT^{2} \)
29 \( 1 - 1.44e6T + 1.45e13T^{2} \)
31 \( 1 + 1.57e6T + 2.64e13T^{2} \)
37 \( 1 + (1.41e7 + 1.41e7i)T + 1.29e14iT^{2} \)
41 \( 1 + 1.52e7iT - 3.27e14T^{2} \)
43 \( 1 + (1.45e7 - 1.45e7i)T - 5.02e14iT^{2} \)
47 \( 1 + (2.49e7 - 2.49e7i)T - 1.11e15iT^{2} \)
53 \( 1 + (2.38e7 + 2.38e7i)T + 3.29e15iT^{2} \)
59 \( 1 + 1.41e8T + 8.66e15T^{2} \)
61 \( 1 - 2.77e7T + 1.16e16T^{2} \)
67 \( 1 + (5.16e7 + 5.16e7i)T + 2.72e16iT^{2} \)
71 \( 1 + 2.50e8iT - 4.58e16T^{2} \)
73 \( 1 + (-8.81e7 + 8.81e7i)T - 5.88e16iT^{2} \)
79 \( 1 + 3.89e8iT - 1.19e17T^{2} \)
83 \( 1 + (9.47e6 + 9.47e6i)T + 1.86e17iT^{2} \)
89 \( 1 - 4.88e8T + 3.50e17T^{2} \)
97 \( 1 + (7.00e7 + 7.00e7i)T + 7.60e17iT^{2} \)
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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

−12.43072408595513262562423427388, −11.12871681791001268678371336323, −10.51656995976314725805970637949, −8.529827941878913382244329087414, −7.39538845335171503527473728902, −6.21789660621840296412034067917, −4.75864856313112164984674773299, −3.41475611099982106522026128923, −1.84634678207708784575610327993, −0.24161129945717239018415563673, 1.36658689287492063801879381930, 3.70472018611425232347737424191, 4.90161487111987670670090269583, 6.07513551814724745151501941493, 6.68888079620289409577796391295, 8.852743901908296818185789530131, 9.874370911779607071303223082583, 10.99647811777532838501686490836, 11.88113617588189535823957147305, 13.37031109354449465528453526820

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