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Properties

Label 2-475-19.18-c2-0-23
Degree $2$
Conductor $475$
Sign $-0.157 - 0.987i$
Analytic cond. $12.9428$
Root an. cond. $3.59761$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.151i·2-s + 5.10i·3-s + 3.97·4-s + 0.774·6-s + 6.35·7-s − 1.20i·8-s − 17.0·9-s + 10.0·11-s + 20.2i·12-s + 3.79i·13-s − 0.963i·14-s + 15.7·16-s + 13.4·17-s + 2.58i·18-s + (2.99 + 18.7i)19-s + ⋯
L(s)  = 1  − 0.0758i·2-s + 1.70i·3-s + 0.994·4-s + 0.129·6-s + 0.907·7-s − 0.151i·8-s − 1.89·9-s + 0.913·11-s + 1.69i·12-s + 0.291i·13-s − 0.0688i·14-s + 0.982·16-s + 0.790·17-s + 0.143i·18-s + (0.157 + 0.987i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.157 - 0.987i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.157 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.157 - 0.987i$
Analytic conductor: \(12.9428\)
Root analytic conductor: \(3.59761\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1),\ -0.157 - 0.987i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.487972969\)
\(L(\frac12)\) \(\approx\) \(2.487972969\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 + (-2.99 - 18.7i)T \)
good2 \( 1 + 0.151iT - 4T^{2} \)
3 \( 1 - 5.10iT - 9T^{2} \)
7 \( 1 - 6.35T + 49T^{2} \)
11 \( 1 - 10.0T + 121T^{2} \)
13 \( 1 - 3.79iT - 169T^{2} \)
17 \( 1 - 13.4T + 289T^{2} \)
23 \( 1 + 16.7T + 529T^{2} \)
29 \( 1 + 24.9iT - 841T^{2} \)
31 \( 1 - 46.3iT - 961T^{2} \)
37 \( 1 + 68.4iT - 1.36e3T^{2} \)
41 \( 1 + 47.5iT - 1.68e3T^{2} \)
43 \( 1 + 43.4T + 1.84e3T^{2} \)
47 \( 1 + 37.4T + 2.20e3T^{2} \)
53 \( 1 - 9.48iT - 2.80e3T^{2} \)
59 \( 1 - 87.7iT - 3.48e3T^{2} \)
61 \( 1 + 95.9T + 3.72e3T^{2} \)
67 \( 1 - 75.1iT - 4.48e3T^{2} \)
71 \( 1 + 77.6iT - 5.04e3T^{2} \)
73 \( 1 - 120.T + 5.32e3T^{2} \)
79 \( 1 + 12.8iT - 6.24e3T^{2} \)
83 \( 1 - 108.T + 6.88e3T^{2} \)
89 \( 1 + 156. iT - 7.92e3T^{2} \)
97 \( 1 + 56.1iT - 9.40e3T^{2} \)
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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

−10.85531304204087051101536341956, −10.31235966329640504861657953465, −9.469083198626013001975319774030, −8.490558602118416179876059663104, −7.50804588135305102101603507213, −6.14453014196730027865756113508, −5.29043368480842507648955618365, −4.13440305496327930520104874931, −3.34726514931457442340121872434, −1.75378823280899979374114101782, 1.12581357508599199782770249869, 1.94245944784649166342110320115, 3.16663995186180601286331057514, 5.10666953606164657698477723842, 6.32663659668917800804555379489, 6.75472527376347850221531913455, 7.963436203198554387698522612225, 8.051892320964799229447267310621, 9.613680969342359342297650818835, 11.00574425951582098853963078761

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