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Properties

Label 1-1625-1625.654-r0-0-0
Degree $1$
Conductor $1625$
Sign $0.699 - 0.714i$
Analytic cond. $7.54646$
Root an. cond. $7.54646$
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  + (0.228 − 0.973i)2-s + (0.999 + 0.0418i)3-s + (−0.895 − 0.444i)4-s + (0.268 − 0.963i)6-s + (−0.978 − 0.207i)7-s + (−0.637 + 0.770i)8-s + (0.996 + 0.0836i)9-s + (−0.228 + 0.973i)11-s + (−0.876 − 0.481i)12-s + (−0.425 + 0.904i)14-s + (0.604 + 0.796i)16-s + (−0.832 − 0.553i)17-s + (0.309 − 0.951i)18-s + (0.999 − 0.0418i)19-s + (−0.968 − 0.248i)21-s + (0.895 + 0.444i)22-s + ⋯
L(s)  = 1  + (0.228 − 0.973i)2-s + (0.999 + 0.0418i)3-s + (−0.895 − 0.444i)4-s + (0.268 − 0.963i)6-s + (−0.978 − 0.207i)7-s + (−0.637 + 0.770i)8-s + (0.996 + 0.0836i)9-s + (−0.228 + 0.973i)11-s + (−0.876 − 0.481i)12-s + (−0.425 + 0.904i)14-s + (0.604 + 0.796i)16-s + (−0.832 − 0.553i)17-s + (0.309 − 0.951i)18-s + (0.999 − 0.0418i)19-s + (−0.968 − 0.248i)21-s + (0.895 + 0.444i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1625 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.699 - 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1625 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.699 - 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1625\)    =    \(5^{3} \cdot 13\)
Sign: $0.699 - 0.714i$
Analytic conductor: \(7.54646\)
Root analytic conductor: \(7.54646\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1625} (654, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1625,\ (0:\ ),\ 0.699 - 0.714i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.859955190 - 0.7815716131i\)
\(L(\frac12)\) \(\approx\) \(1.859955190 - 0.7815716131i\)
\(L(1)\) \(\approx\) \(1.270654241 - 0.5501067556i\)
\(L(1)\) \(\approx\) \(1.270654241 - 0.5501067556i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.228 - 0.973i)T \)
3 \( 1 + (0.999 + 0.0418i)T \)
7 \( 1 + (-0.978 - 0.207i)T \)
11 \( 1 + (-0.228 + 0.973i)T \)
17 \( 1 + (-0.832 - 0.553i)T \)
19 \( 1 + (0.999 - 0.0418i)T \)
23 \( 1 + (0.756 - 0.653i)T \)
29 \( 1 + (0.783 + 0.621i)T \)
31 \( 1 + (-0.0627 + 0.998i)T \)
37 \( 1 + (0.604 + 0.796i)T \)
41 \( 1 + (-0.944 + 0.328i)T \)
43 \( 1 + (0.104 + 0.994i)T \)
47 \( 1 + (-0.637 - 0.770i)T \)
53 \( 1 + (-0.968 - 0.248i)T \)
59 \( 1 + (0.855 - 0.518i)T \)
61 \( 1 + (0.944 + 0.328i)T \)
67 \( 1 + (0.146 + 0.989i)T \)
71 \( 1 + (-0.985 + 0.166i)T \)
73 \( 1 + (0.876 - 0.481i)T \)
79 \( 1 + (0.535 - 0.844i)T \)
83 \( 1 + (0.535 + 0.844i)T \)
89 \( 1 + (0.855 + 0.518i)T \)
97 \( 1 + (0.146 - 0.989i)T \)
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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

−20.57166349428479539634440499284, −19.560279732360269371770192088736, −19.03821374110269502068834006372, −18.40247632333521122448731854644, −17.47518191397225924763653284029, −16.57699392996109246162088326268, −15.74520078931766151494090093057, −15.53041239466189574223764760584, −14.572573671440125621734092416305, −13.73529164322226586337501075774, −13.29575335202800212581332351889, −12.705799412416380409099213489069, −11.62206763323597814203861558371, −10.35012649289506714193457507631, −9.45712464720611636238570003389, −8.98990573768956988838562529387, −8.17481719122914682541747492321, −7.461875760498603680387393382627, −6.59785458430212373872915538044, −5.92248978254655218242723799303, −4.92148569243280414877746452555, −3.79701603019745938476276773383, −3.315586142583113016521452315618, −2.37986235103934598304616094415, −0.748732450993837745932581867828, 0.97903540545930482122135368373, 2.049641041354251133694625739490, 2.94629234224858179197165374395, 3.3606407995917203895443682664, 4.53646608833241911796306403891, 4.9957261514189738532863063812, 6.543640558856364927622457905423, 7.1724390230638205903083737761, 8.34584545608599763648249905821, 9.05647352454178125238625827070, 9.85080621601628247976118497529, 10.15206671299895129618941745509, 11.244422063287554330884986305093, 12.24251620466676851650801417970, 12.92123502597412356232129723195, 13.39684449108703742863059599370, 14.18926495090424443130754973291, 14.92740663922698700734318559587, 15.699643398592737908470201751595, 16.478830610272639885465422726707, 17.84428720657220789237405964826, 18.2222758579289083054302901674, 19.20833542425263204482198496862, 19.68131005695228395383105773456, 20.45064208240142652748342821526

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