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Properties

Label 1-475-475.187-r1-0-0
Degree $1$
Conductor $475$
Sign $-0.791 - 0.611i$
Analytic cond. $51.0458$
Root an. cond. $51.0458$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.469 − 0.882i)2-s + (0.970 + 0.241i)3-s + (−0.559 + 0.829i)4-s + (−0.241 − 0.970i)6-s + (−0.866 − 0.5i)7-s + (0.994 + 0.104i)8-s + (0.882 + 0.469i)9-s + (0.669 − 0.743i)11-s + (−0.743 + 0.669i)12-s + (−0.529 − 0.848i)13-s + (−0.0348 + 0.999i)14-s + (−0.374 − 0.927i)16-s + (0.898 + 0.438i)17-s − i·18-s + (−0.719 − 0.694i)21-s + (−0.970 − 0.241i)22-s + ⋯
L(s)  = 1  + (−0.469 − 0.882i)2-s + (0.970 + 0.241i)3-s + (−0.559 + 0.829i)4-s + (−0.241 − 0.970i)6-s + (−0.866 − 0.5i)7-s + (0.994 + 0.104i)8-s + (0.882 + 0.469i)9-s + (0.669 − 0.743i)11-s + (−0.743 + 0.669i)12-s + (−0.529 − 0.848i)13-s + (−0.0348 + 0.999i)14-s + (−0.374 − 0.927i)16-s + (0.898 + 0.438i)17-s − i·18-s + (−0.719 − 0.694i)21-s + (−0.970 − 0.241i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.791 - 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.791 - 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.791 - 0.611i$
Analytic conductor: \(51.0458\)
Root analytic conductor: \(51.0458\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (1:\ ),\ -0.791 - 0.611i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4868598371 - 1.427356287i\)
\(L(\frac12)\) \(\approx\) \(0.4868598371 - 1.427356287i\)
\(L(1)\) \(\approx\) \(0.8997123915 - 0.4895615598i\)
\(L(1)\) \(\approx\) \(0.8997123915 - 0.4895615598i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.469 - 0.882i)T \)
3 \( 1 + (0.970 + 0.241i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (0.669 - 0.743i)T \)
13 \( 1 + (-0.529 - 0.848i)T \)
17 \( 1 + (0.898 + 0.438i)T \)
23 \( 1 + (-0.788 + 0.615i)T \)
29 \( 1 + (-0.438 - 0.898i)T \)
31 \( 1 + (0.913 - 0.406i)T \)
37 \( 1 + (0.951 + 0.309i)T \)
41 \( 1 + (-0.374 - 0.927i)T \)
43 \( 1 + (-0.342 + 0.939i)T \)
47 \( 1 + (-0.898 + 0.438i)T \)
53 \( 1 + (-0.829 - 0.559i)T \)
59 \( 1 + (-0.990 - 0.139i)T \)
61 \( 1 + (-0.615 - 0.788i)T \)
67 \( 1 + (-0.694 - 0.719i)T \)
71 \( 1 + (0.961 - 0.275i)T \)
73 \( 1 + (0.529 - 0.848i)T \)
79 \( 1 + (0.241 - 0.970i)T \)
83 \( 1 + (-0.406 - 0.913i)T \)
89 \( 1 + (0.374 - 0.927i)T \)
97 \( 1 + (0.694 - 0.719i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.227608215766157545107035837, −23.30290822842973516772571679016, −22.38828101988164878985124793154, −21.47730421095586978159420168856, −20.10385209422565092493039586641, −19.62043693894183424140246781321, −18.69845514626203682381695904148, −18.20975367521910759469020050845, −16.897233109802043292218835744349, −16.20679045625091908582982273270, −15.25449335666505834622380313911, −14.52337201521833587677355894767, −13.89065391820636060986831739593, −12.76135315047549190450067756089, −11.949532699871299731685927789715, −10.12130270240117125609111951171, −9.55457011155560322811992220765, −8.89478330405790534616778608077, −7.85396265615366436373562644810, −6.93724028958786785927315180368, −6.32193160619800536688881461017, −4.88709409666144925737468201894, −3.80722059982233756897252561182, −2.451240346379291198133867825838, −1.27601774219798799135675582052, 0.42313089816424608717692290229, 1.6801237227154847739871514228, 3.05106573324059188762269218896, 3.489518664740601152406670904206, 4.53835170135022165152215069835, 6.18024621234906600831359977470, 7.66016247477917042895207159524, 8.120745891400462297886100530834, 9.414487930684911704485854218589, 9.82779345272544038865337872279, 10.68370749279109330291087402186, 11.89203805643777558418918765016, 12.86887788371013947888642396319, 13.56413337044902974843383288640, 14.369745159426983650894627236979, 15.56168558439395533846842123770, 16.5622218149664696066966814101, 17.25128622468353934929119512662, 18.52255177379666510324048207687, 19.356159872821619097874482660, 19.70856999501760549244488541702, 20.549619497879494995823990561650, 21.4243811511142280250942672137, 22.170554582013303897165775427586, 22.9487647070620910737362314427

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