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Properties

Label 1-6025-6025.1359-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.788 - 0.615i$
Analytic cond. $27.9799$
Root an. cond. $27.9799$
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  + (−0.309 − 0.951i)2-s + (−0.309 − 0.951i)3-s + (−0.809 + 0.587i)4-s + (−0.809 + 0.587i)6-s + (−0.809 + 0.587i)7-s + (0.809 + 0.587i)8-s + (−0.809 + 0.587i)9-s + (−0.309 − 0.951i)11-s + (0.809 + 0.587i)12-s + (−0.809 − 0.587i)13-s + (0.809 + 0.587i)14-s + (0.309 − 0.951i)16-s + 17-s + (0.809 + 0.587i)18-s + (0.809 + 0.587i)19-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)2-s + (−0.309 − 0.951i)3-s + (−0.809 + 0.587i)4-s + (−0.809 + 0.587i)6-s + (−0.809 + 0.587i)7-s + (0.809 + 0.587i)8-s + (−0.809 + 0.587i)9-s + (−0.309 − 0.951i)11-s + (0.809 + 0.587i)12-s + (−0.809 − 0.587i)13-s + (0.809 + 0.587i)14-s + (0.309 − 0.951i)16-s + 17-s + (0.809 + 0.587i)18-s + (0.809 + 0.587i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.788 - 0.615i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (1359, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.788 - 0.615i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6413883011 - 0.2205899220i\)
\(L(\frac12)\) \(\approx\) \(0.6413883011 - 0.2205899220i\)
\(L(1)\) \(\approx\) \(0.5220186468 - 0.3785465049i\)
\(L(1)\) \(\approx\) \(0.5220186468 - 0.3785465049i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.309 - 0.951i)T \)
3 \( 1 + (-0.309 - 0.951i)T \)
7 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (-0.309 - 0.951i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
17 \( 1 + T \)
19 \( 1 + (0.809 + 0.587i)T \)
23 \( 1 + T \)
29 \( 1 + T \)
31 \( 1 - T \)
37 \( 1 + T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + (-0.809 - 0.587i)T \)
47 \( 1 - T \)
53 \( 1 - T \)
59 \( 1 + T \)
61 \( 1 + T \)
67 \( 1 - T \)
71 \( 1 - T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (-0.309 + 0.951i)T \)
89 \( 1 + (-0.309 - 0.951i)T \)
97 \( 1 - T \)
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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

−17.57329045396543078409297521041, −16.97105420765666467047970498929, −16.35620811463933854627811912329, −16.05069211008613752630264157595, −15.24644695883902856556563389053, −14.62552406624397836476189043333, −14.23115917127826689501158028580, −13.25331150497771399933296643647, −12.66001595193587546197399743283, −11.79514464863227736338846716647, −10.935043914843070250613197470456, −10.09672464888218375975389304340, −9.78766663851203523403190309405, −9.321409558665388463306319797232, −8.51154305328417252897081734961, −7.4187375065441852027051361638, −7.147441543365413304869322760354, −6.3353062179596786056549653660, −5.529286387016490915340691695, −4.837395169903272754309650798404, −4.42920247527888234681349338088, −3.49060354416928563505319016070, −2.763016424807120497143275462920, −1.34628663188225803212333199543, −0.31522633671080407330095071380, 0.70132386876657969275710079643, 1.36478082640849922760647937398, 2.41366351258428545962702723605, 3.09682820792259290295918573176, 3.32922309659460033831018654085, 4.800310481617906136609276559, 5.4785282239906723002729094099, 5.97993946626134072851350215606, 7.03595401141519678544837563775, 7.701270623903429100549546184621, 8.314703087197293093692499567970, 8.96138615450585113788277818819, 9.880450440956441372980184673874, 10.25433383482928508281684397695, 11.29819982620538364396305044244, 11.67703904994025615000554267493, 12.393341737516003980983225847663, 12.95662062121639549283110797006, 13.26598523248821316002193238942, 14.25523387153157292202042766992, 14.706817834766909497781532184837, 16.015990963382567898097149305664, 16.53447922364017081434623917457, 17.0586102004593399396244371399, 18.001973691917024560837779555146

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