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Properties

Label 1-5e4-625.29-r0-0-0
Degree $1$
Conductor $625$
Sign $-0.434 + 0.900i$
Analytic cond. $2.90248$
Root an. cond. $2.90248$
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  + (−0.448 + 0.893i)2-s + (−0.492 − 0.870i)3-s + (−0.597 − 0.801i)4-s + (0.998 − 0.0502i)6-s + (0.637 + 0.770i)7-s + (0.984 − 0.175i)8-s + (−0.514 + 0.857i)9-s + (0.162 + 0.986i)11-s + (−0.402 + 0.915i)12-s + (0.974 + 0.224i)13-s + (−0.974 + 0.224i)14-s + (−0.285 + 0.958i)16-s + (0.947 + 0.320i)17-s + (−0.535 − 0.844i)18-s + (−0.910 + 0.414i)19-s + ⋯
L(s)  = 1  + (−0.448 + 0.893i)2-s + (−0.492 − 0.870i)3-s + (−0.597 − 0.801i)4-s + (0.998 − 0.0502i)6-s + (0.637 + 0.770i)7-s + (0.984 − 0.175i)8-s + (−0.514 + 0.857i)9-s + (0.162 + 0.986i)11-s + (−0.402 + 0.915i)12-s + (0.974 + 0.224i)13-s + (−0.974 + 0.224i)14-s + (−0.285 + 0.958i)16-s + (0.947 + 0.320i)17-s + (−0.535 − 0.844i)18-s + (−0.910 + 0.414i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(625\)    =    \(5^{4}\)
Sign: $-0.434 + 0.900i$
Analytic conductor: \(2.90248\)
Root analytic conductor: \(2.90248\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{625} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 625,\ (0:\ ),\ -0.434 + 0.900i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3958734295 + 0.6307828061i\)
\(L(\frac12)\) \(\approx\) \(0.3958734295 + 0.6307828061i\)
\(L(1)\) \(\approx\) \(0.6568926571 + 0.2756842350i\)
\(L(1)\) \(\approx\) \(0.6568926571 + 0.2756842350i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
good2 \( 1 + (-0.448 + 0.893i)T \)
3 \( 1 + (-0.492 - 0.870i)T \)
7 \( 1 + (0.637 + 0.770i)T \)
11 \( 1 + (0.162 + 0.986i)T \)
13 \( 1 + (0.974 + 0.224i)T \)
17 \( 1 + (0.947 + 0.320i)T \)
19 \( 1 + (-0.910 + 0.414i)T \)
23 \( 1 + (-0.617 - 0.786i)T \)
29 \( 1 + (-0.997 + 0.0753i)T \)
31 \( 1 + (-0.947 - 0.320i)T \)
37 \( 1 + (-0.823 - 0.567i)T \)
41 \( 1 + (-0.0376 + 0.999i)T \)
43 \( 1 + (0.425 + 0.904i)T \)
47 \( 1 + (-0.899 + 0.437i)T \)
53 \( 1 + (0.837 - 0.546i)T \)
59 \( 1 + (-0.863 + 0.503i)T \)
61 \( 1 + (-0.0376 - 0.999i)T \)
67 \( 1 + (0.236 + 0.971i)T \)
71 \( 1 + (0.693 + 0.720i)T \)
73 \( 1 + (-0.212 + 0.977i)T \)
79 \( 1 + (0.492 + 0.870i)T \)
83 \( 1 + (-0.112 - 0.993i)T \)
89 \( 1 + (0.402 + 0.915i)T \)
97 \( 1 + (0.379 + 0.925i)T \)
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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

−22.51587824924796120006322191168, −21.63409448370374325148855209705, −21.02842198434187715508890704796, −20.47352054588592067453505381288, −19.55105653148411805502262773391, −18.5382942810428850149511631025, −17.76419009720988802783245847574, −16.8785338936047544847965783591, −16.463910541009193487568169328188, −15.325969578150715492624201273, −14.116503557150507946832885257667, −13.50864098533129566623712956078, −12.258262751458893145000932726033, −11.35495429491309740638084140552, −10.84114514717224992919815202339, −10.20502813987591863691250879348, −9.10054730424489452608211446753, −8.41941370195567023396179376430, −7.34195242070612886130184779823, −5.88679293209851706335206543190, −4.945275442944627691489467756861, −3.75665154116581449694248917239, −3.44497302303548956735486008981, −1.693022017469012231135792149151, −0.50359038687906694421324798305, 1.385308068001953070661662806985, 2.07760496526450362949037250809, 4.1040461541380089066315592762, 5.235583208274910209494025819699, 5.967394007644245369283778533755, 6.72243290586989352475684106022, 7.80147208897512528880054601191, 8.31949500992158733651958508550, 9.33488996562530744407818047375, 10.50172628070453829250613456054, 11.35672948983505695714046852347, 12.44230004522423451268620065463, 13.08982076407327365063451254096, 14.42066466099979362160425562994, 14.71162876149878233676098796334, 15.93508902659950540826920885393, 16.73881497837957695601017371675, 17.49564523767142597729868003516, 18.356791455660298359622959145222, 18.61289447142789777885014880356, 19.61698773193340474795354053730, 20.70553843646184744649212947513, 21.82328021654643923887883087850, 22.90985286688913880685154644730, 23.26171231224400880856478187290

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