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Properties

Label 2-475-19.5-c1-0-5
Degree $2$
Conductor $475$
Sign $0.0540 - 0.998i$
Analytic cond. $3.79289$
Root an. cond. $1.94753$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.826 − 0.300i)2-s + (−0.0923 + 0.524i)3-s + (−0.939 + 0.788i)4-s + (0.0812 + 0.460i)6-s + (−0.939 − 1.62i)7-s + (−1.41 + 2.45i)8-s + (2.55 + 0.929i)9-s + (−1.70 + 2.95i)11-s + (−0.326 − 0.565i)12-s + (0.918 + 5.21i)13-s + (−1.26 − 1.06i)14-s + (−0.00727 + 0.0412i)16-s + (1.55 − 0.565i)17-s + 2.38·18-s + (−2.52 + 3.55i)19-s + ⋯
L(s)  = 1  + (0.584 − 0.212i)2-s + (−0.0533 + 0.302i)3-s + (−0.469 + 0.394i)4-s + (0.0331 + 0.188i)6-s + (−0.355 − 0.615i)7-s + (−0.501 + 0.868i)8-s + (0.851 + 0.309i)9-s + (−0.514 + 0.890i)11-s + (−0.0942 − 0.163i)12-s + (0.254 + 1.44i)13-s + (−0.338 − 0.283i)14-s + (−0.00181 + 0.0103i)16-s + (0.376 − 0.137i)17-s + 0.563·18-s + (−0.578 + 0.815i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0540 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0540 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.0540 - 0.998i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (176, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.0540 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01457 + 0.961089i\)
\(L(\frac12)\) \(\approx\) \(1.01457 + 0.961089i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.52 - 3.55i)T \)
good2 \( 1 + (-0.826 + 0.300i)T + (1.53 - 1.28i)T^{2} \)
3 \( 1 + (0.0923 - 0.524i)T + (-2.81 - 1.02i)T^{2} \)
7 \( 1 + (0.939 + 1.62i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.70 - 2.95i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.918 - 5.21i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (-1.55 + 0.565i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (1.34 - 1.13i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-3.25 - 1.18i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (0.971 + 1.68i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 0.837T + 37T^{2} \)
41 \( 1 + (0.779 - 4.42i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (3.67 + 3.08i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (-0.673 - 0.245i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (-4.67 + 3.92i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (-10.1 + 3.67i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-3.36 + 2.82i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-13.3 - 4.86i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (10.5 + 8.84i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (-1.30 + 7.40i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (1.20 - 6.85i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-1.25 - 2.17i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.396 + 2.24i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (1.71 - 0.623i)T + (74.3 - 62.3i)T^{2} \)
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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

−11.35809213319122340214395064551, −10.18882750767753417584647006017, −9.689779395306242880868421784007, −8.541972931398374095547550251566, −7.53023384693128503290569106898, −6.63435142899018125629169885324, −5.19322245987268443462073409156, −4.31130249934668474491845248528, −3.70229199809867537929141549211, −2.03581451027914433394361776738, 0.74613215624679572340325425385, 2.85076074105762926902244663881, 4.00381889577203808168250445722, 5.27638082088564376033295735820, 5.93572076229043100699635684112, 6.84127753243863341222283845562, 8.127942241661077722077818922605, 8.943280612223013339601956439181, 10.02502390476294829183256548518, 10.61432525351406471804625193223

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