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Properties

Label 2-75-5.4-c9-0-13
Degree $2$
Conductor $75$
Sign $-0.894 + 0.447i$
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  − 43.8i·2-s + 81i·3-s − 1.41e3·4-s + 3.55e3·6-s + 7.86e3i·7-s + 3.95e4i·8-s − 6.56e3·9-s − 4.93e4·11-s − 1.14e5i·12-s + 2.42e4i·13-s + 3.44e5·14-s + 1.01e6·16-s − 2.68e5i·17-s + 2.87e5i·18-s + 1.68e5·19-s + ⋯
L(s)  = 1  − 1.93i·2-s + 0.577i·3-s − 2.76·4-s + 1.11·6-s + 1.23i·7-s + 3.41i·8-s − 0.333·9-s − 1.01·11-s − 1.59i·12-s + 0.235i·13-s + 2.40·14-s + 3.86·16-s − 0.778i·17-s + 0.646i·18-s + 0.296·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(38.6276\)
Root analytic conductor: \(6.21511\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :9/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.227562 - 0.963969i\)
\(L(\frac12)\) \(\approx\) \(0.227562 - 0.963969i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 81iT \)
5 \( 1 \)
good2 \( 1 + 43.8iT - 512T^{2} \)
7 \( 1 - 7.86e3iT - 4.03e7T^{2} \)
11 \( 1 + 4.93e4T + 2.35e9T^{2} \)
13 \( 1 - 2.42e4iT - 1.06e10T^{2} \)
17 \( 1 + 2.68e5iT - 1.18e11T^{2} \)
19 \( 1 - 1.68e5T + 3.22e11T^{2} \)
23 \( 1 + 2.12e6iT - 1.80e12T^{2} \)
29 \( 1 + 3.89e5T + 1.45e13T^{2} \)
31 \( 1 - 9.05e4T + 2.64e13T^{2} \)
37 \( 1 - 3.31e6iT - 1.29e14T^{2} \)
41 \( 1 - 2.32e7T + 3.27e14T^{2} \)
43 \( 1 - 1.91e7iT - 5.02e14T^{2} \)
47 \( 1 + 6.28e7iT - 1.11e15T^{2} \)
53 \( 1 + 1.80e5iT - 3.29e15T^{2} \)
59 \( 1 + 3.84e7T + 8.66e15T^{2} \)
61 \( 1 + 5.53e5T + 1.16e16T^{2} \)
67 \( 1 - 2.39e8iT - 2.72e16T^{2} \)
71 \( 1 - 1.28e8T + 4.58e16T^{2} \)
73 \( 1 + 2.39e8iT - 5.88e16T^{2} \)
79 \( 1 - 5.28e8T + 1.19e17T^{2} \)
83 \( 1 - 2.12e8iT - 1.86e17T^{2} \)
89 \( 1 - 2.07e8T + 3.50e17T^{2} \)
97 \( 1 + 1.70e9iT - 7.60e17T^{2} \)
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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

−12.02859448982627542427405027037, −11.12323761215009038744536944588, −10.14208660963074776532113234916, −9.210045311509030751356798678809, −8.328464809098468629459041243860, −5.48817746286961414954993311600, −4.53243215947305278474614437816, −2.98786495756296376695435019039, −2.24012796943732581602349303239, −0.39868963713341195395434252827, 0.876751785216445830206569210518, 3.77545806245053456375930399975, 5.17485839969695067942464196411, 6.27049299543984017842672508486, 7.54361226557208892906640073637, 7.82595793625167115516015003601, 9.334885198811853896593316357721, 10.61573746983270781112837074056, 12.74365850126351331239520561159, 13.49365010302607179134430512877

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