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Properties

Label 2-475-95.27-c1-0-8
Degree $2$
Conductor $475$
Sign $0.941 - 0.336i$
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  + (0.258 + 0.965i)2-s + (−3.08 + 0.826i)3-s + (0.866 − 0.5i)4-s + (−1.59 − 2.76i)6-s + (−1.22 − 1.22i)7-s + (2.12 + 2.12i)8-s + (6.22 − 3.59i)9-s − 4.19·11-s + (−2.25 + 2.25i)12-s + (1.08 − 4.04i)13-s + (0.866 − 1.49i)14-s + (−0.500 + 0.866i)16-s + (5.34 − 1.43i)17-s + (5.08 + 5.08i)18-s + (4.33 − 0.5i)19-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−1.78 + 0.477i)3-s + (0.433 − 0.250i)4-s + (−0.651 − 1.12i)6-s + (−0.462 − 0.462i)7-s + (0.749 + 0.749i)8-s + (2.07 − 1.19i)9-s − 1.26·11-s + (−0.651 + 0.651i)12-s + (0.300 − 1.12i)13-s + (0.231 − 0.400i)14-s + (−0.125 + 0.216i)16-s + (1.29 − 0.347i)17-s + (1.19 + 1.19i)18-s + (0.993 − 0.114i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 - 0.336i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.941 - 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.941 - 0.336i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.941 - 0.336i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.956230 + 0.165870i\)
\(L(\frac12)\) \(\approx\) \(0.956230 + 0.165870i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.33 + 0.5i)T \)
good2 \( 1 + (-0.258 - 0.965i)T + (-1.73 + i)T^{2} \)
3 \( 1 + (3.08 - 0.826i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (1.22 + 1.22i)T + 7iT^{2} \)
11 \( 1 + 4.19T + 11T^{2} \)
13 \( 1 + (-1.08 + 4.04i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (-5.34 + 1.43i)T + (14.7 - 8.5i)T^{2} \)
23 \( 1 + (-1.67 - 0.448i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-2.76 - 4.78i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.06iT - 31T^{2} \)
37 \( 1 + (-1.68 + 1.68i)T - 37iT^{2} \)
41 \( 1 + (-3.28 - 1.89i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.0863 - 0.322i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-2.41 + 9.00i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.81 - 6.76i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-4.66 + 8.07i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.28 + 10.8i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-10.4 - 2.78i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (2.07 + 1.19i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.22 + 12.0i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-5.69 + 9.86i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.13 - 5.13i)T - 83iT^{2} \)
89 \( 1 + (2.59 + 4.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.86 - 6.94i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.80887940304836135348628404677, −10.46872560849635341041673999638, −9.739139330349959035256670245546, −7.85804043305020541878201966008, −7.16369492417223030946769227340, −6.20861276975114402339716827205, −5.31182557362519177186111979705, −5.07475901302384567450314313329, −3.29475258960906443161339279776, −0.848698950663153652563095929179, 1.20643862724860589082556342740, 2.67420155311283395551707290707, 4.23437447287638938757221487956, 5.43908822593186205305356109875, 6.16072413476923898344801166610, 7.13679163457241219306723598455, 7.88538700105124755055477158214, 9.720273334879059972464769002903, 10.37378212884283471741817262140, 11.24727505824253801096910383847

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