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Properties

Label 1-475-475.241-r1-0-0
Degree $1$
Conductor $475$
Sign $-0.728 + 0.685i$
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  + (−0.990 + 0.139i)2-s + (0.997 + 0.0697i)3-s + (0.961 − 0.275i)4-s + (−0.997 + 0.0697i)6-s + (−0.5 − 0.866i)7-s + (−0.913 + 0.406i)8-s + (0.990 + 0.139i)9-s + (−0.978 − 0.207i)11-s + (0.978 − 0.207i)12-s + (0.374 + 0.927i)13-s + (0.615 + 0.788i)14-s + (0.848 − 0.529i)16-s + (−0.719 + 0.694i)17-s − 18-s + (−0.438 − 0.898i)21-s + (0.997 + 0.0697i)22-s + ⋯
L(s)  = 1  + (−0.990 + 0.139i)2-s + (0.997 + 0.0697i)3-s + (0.961 − 0.275i)4-s + (−0.997 + 0.0697i)6-s + (−0.5 − 0.866i)7-s + (−0.913 + 0.406i)8-s + (0.990 + 0.139i)9-s + (−0.978 − 0.207i)11-s + (0.978 − 0.207i)12-s + (0.374 + 0.927i)13-s + (0.615 + 0.788i)14-s + (0.848 − 0.529i)16-s + (−0.719 + 0.694i)17-s − 18-s + (−0.438 − 0.898i)21-s + (0.997 + 0.0697i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.728 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.728 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.728 + 0.685i$
Analytic conductor: \(51.0458\)
Root analytic conductor: \(51.0458\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (1:\ ),\ -0.728 + 0.685i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2190793911 + 0.5528042221i\)
\(L(\frac12)\) \(\approx\) \(0.2190793911 + 0.5528042221i\)
\(L(1)\) \(\approx\) \(0.7711693195 + 0.08164896111i\)
\(L(1)\) \(\approx\) \(0.7711693195 + 0.08164896111i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.990 + 0.139i)T \)
3 \( 1 + (0.997 + 0.0697i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.978 - 0.207i)T \)
13 \( 1 + (0.374 + 0.927i)T \)
17 \( 1 + (-0.719 + 0.694i)T \)
23 \( 1 + (0.0348 - 0.999i)T \)
29 \( 1 + (0.719 + 0.694i)T \)
31 \( 1 + (0.104 - 0.994i)T \)
37 \( 1 + (-0.309 + 0.951i)T \)
41 \( 1 + (-0.848 + 0.529i)T \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (-0.719 - 0.694i)T \)
53 \( 1 + (-0.961 + 0.275i)T \)
59 \( 1 + (0.882 + 0.469i)T \)
61 \( 1 + (0.0348 - 0.999i)T \)
67 \( 1 + (-0.438 + 0.898i)T \)
71 \( 1 + (-0.559 + 0.829i)T \)
73 \( 1 + (-0.374 + 0.927i)T \)
79 \( 1 + (0.997 + 0.0697i)T \)
83 \( 1 + (-0.104 + 0.994i)T \)
89 \( 1 + (-0.848 - 0.529i)T \)
97 \( 1 + (-0.438 - 0.898i)T \)
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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

−23.49377714609794907303726689013, −22.191788621886864276107670764581, −21.19512875570241332802603619625, −20.63833363075785469069303555562, −19.68125546596141183353467898449, −19.13748885082250402285235981680, −18.10492730227370265794186944709, −17.79675567034596095869418755230, −16.07843841368431645354956952023, −15.660168362700093888579382701702, −15.03130896906539607275134379984, −13.574161768023971803481247745040, −12.80184544354894134562183786200, −11.898482152470297774730812538378, −10.63065854224059845371248083959, −9.84610919937825151701741729828, −9.01116393369358365329728381181, −8.27421583864395227305290816675, −7.46299533075108318761792550235, −6.44082465193347976798184823150, −5.165552695944914543524897918506, −3.37265577125503777875394100610, −2.73292535433380891044603922805, −1.76155175504433015629515508894, −0.17835871607981982612982074164, 1.30321455641491358207455932004, 2.42428512400322627599712471181, 3.43878328717051794559579613811, 4.64776526456680169599137531347, 6.41141599635463506998039639406, 7.016262635199698934582804087029, 8.16498651693080696378415239632, 8.65159320585480473872295450829, 9.83379281980562888195661249552, 10.35962033462161334313603642562, 11.333764072365408233084356153785, 12.78459339522626218123072033227, 13.55703090935515206047693567080, 14.552100261647335612078762920190, 15.48872201965203886192948758614, 16.23653947175719815784511941642, 16.95396172967983066720457235728, 18.21004518198120178334139571326, 18.84302688592083576174455914886, 19.62478386173367796517364121031, 20.36977093306920934141166233308, 20.98680043623489871350335410806, 21.97286423169341630602721968898, 23.575691212141607375919122968580, 23.98363001539581734289034806243

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