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Properties

Label 1-475-475.42-r1-0-0
Degree $1$
Conductor $475$
Sign $-0.815 - 0.578i$
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  + (0.0697 − 0.997i)2-s + (0.999 − 0.0348i)3-s + (−0.990 − 0.139i)4-s + (0.0348 − 0.999i)6-s + (0.866 − 0.5i)7-s + (−0.207 + 0.978i)8-s + (0.997 − 0.0697i)9-s + (−0.104 − 0.994i)11-s + (−0.994 − 0.104i)12-s + (0.829 − 0.559i)13-s + (−0.438 − 0.898i)14-s + (0.961 + 0.275i)16-s + (−0.927 − 0.374i)17-s − i·18-s + (0.848 − 0.529i)21-s + (−0.999 + 0.0348i)22-s + ⋯
L(s)  = 1  + (0.0697 − 0.997i)2-s + (0.999 − 0.0348i)3-s + (−0.990 − 0.139i)4-s + (0.0348 − 0.999i)6-s + (0.866 − 0.5i)7-s + (−0.207 + 0.978i)8-s + (0.997 − 0.0697i)9-s + (−0.104 − 0.994i)11-s + (−0.994 − 0.104i)12-s + (0.829 − 0.559i)13-s + (−0.438 − 0.898i)14-s + (0.961 + 0.275i)16-s + (−0.927 − 0.374i)17-s − i·18-s + (0.848 − 0.529i)21-s + (−0.999 + 0.0348i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.815 - 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.815 - 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.815 - 0.578i$
Analytic conductor: \(51.0458\)
Root analytic conductor: \(51.0458\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (42, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (1:\ ),\ -0.815 - 0.578i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9191252120 - 2.882865014i\)
\(L(\frac12)\) \(\approx\) \(0.9191252120 - 2.882865014i\)
\(L(1)\) \(\approx\) \(1.201125735 - 1.050671574i\)
\(L(1)\) \(\approx\) \(1.201125735 - 1.050671574i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.0697 - 0.997i)T \)
3 \( 1 + (0.999 - 0.0348i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (-0.104 - 0.994i)T \)
13 \( 1 + (0.829 - 0.559i)T \)
17 \( 1 + (-0.927 - 0.374i)T \)
23 \( 1 + (-0.694 + 0.719i)T \)
29 \( 1 + (0.374 + 0.927i)T \)
31 \( 1 + (0.669 - 0.743i)T \)
37 \( 1 + (0.587 - 0.809i)T \)
41 \( 1 + (0.961 + 0.275i)T \)
43 \( 1 + (-0.984 - 0.173i)T \)
47 \( 1 + (0.927 - 0.374i)T \)
53 \( 1 + (0.139 - 0.990i)T \)
59 \( 1 + (0.241 + 0.970i)T \)
61 \( 1 + (-0.719 - 0.694i)T \)
67 \( 1 + (-0.529 + 0.848i)T \)
71 \( 1 + (-0.882 - 0.469i)T \)
73 \( 1 + (-0.829 - 0.559i)T \)
79 \( 1 + (-0.0348 - 0.999i)T \)
83 \( 1 + (-0.743 - 0.669i)T \)
89 \( 1 + (-0.961 + 0.275i)T \)
97 \( 1 + (0.529 + 0.848i)T \)
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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

−24.13997922759269310728108545141, −23.39829259133095479771012877975, −22.26792877309353090703122054468, −21.401655235129055949422440319071, −20.6827700217899728515218871005, −19.64672297836276719382980432898, −18.54191301170584614953620599953, −18.05410044663892528580269691647, −17.10163505548594416054819358495, −15.776028283328352034181562762969, −15.39061852359165560296564476712, −14.497197962626194641000899063949, −13.84733387069594104241023947500, −12.9465264456164648755692057528, −11.960277543469401089645450528719, −10.474187915806257552518233734353, −9.43202876509922170374168215609, −8.584764491129228589156982026594, −8.04436061325571974783380965094, −7.00240530970230516838977263040, −6.05939022702697639968654012031, −4.56643242416105148463243123949, −4.223289071445945183355008118345, −2.60360461091378733739151972359, −1.43915256683439019978330396710, 0.70794408280247783140517177114, 1.71970073396714227566706836634, 2.835133319541742725909505954772, 3.76008175149350705998440487384, 4.58946017454204730514747036392, 5.86944558258648301157340802189, 7.5018972130716016755342498852, 8.38716053141448957787857001109, 8.95652908075488506413957853857, 10.15305227610833432904810165677, 10.93061328480001975584843544339, 11.72479978253967544252174105445, 13.087585240597138311425744169, 13.59436777233092847321765982858, 14.26405445905475939101017298462, 15.24042099391796585491560651557, 16.328143080624174228532656380005, 17.78515603135813471866357739729, 18.21367983160216089679579682349, 19.25306156542890863376083319430, 20.0145127242734513496729260937, 20.61674447027738870547550386227, 21.36165772240559943341478191740, 22.056603388050090342473422694456, 23.32027441495501930628282449479

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