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Properties

Label 1-6025-6025.244-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.876 - 0.482i$
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  + (0.994 − 0.104i)2-s + (−0.406 + 0.913i)3-s + (0.978 − 0.207i)4-s + (−0.309 + 0.951i)6-s + (−0.0523 + 0.998i)7-s + (0.951 − 0.309i)8-s + (−0.669 − 0.743i)9-s + (−0.629 + 0.777i)11-s + (−0.207 + 0.978i)12-s + (−0.0523 − 0.998i)13-s + (0.0523 + 0.998i)14-s + (0.913 − 0.406i)16-s + (0.707 − 0.707i)17-s + (−0.743 − 0.669i)18-s + (−0.544 − 0.838i)19-s + ⋯
L(s)  = 1  + (0.994 − 0.104i)2-s + (−0.406 + 0.913i)3-s + (0.978 − 0.207i)4-s + (−0.309 + 0.951i)6-s + (−0.0523 + 0.998i)7-s + (0.951 − 0.309i)8-s + (−0.669 − 0.743i)9-s + (−0.629 + 0.777i)11-s + (−0.207 + 0.978i)12-s + (−0.0523 − 0.998i)13-s + (0.0523 + 0.998i)14-s + (0.913 − 0.406i)16-s + (0.707 − 0.707i)17-s + (−0.743 − 0.669i)18-s + (−0.544 − 0.838i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.482i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.482i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.876 - 0.482i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.876 - 0.482i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.346281882 - 0.6030802586i\)
\(L(\frac12)\) \(\approx\) \(2.346281882 - 0.6030802586i\)
\(L(1)\) \(\approx\) \(1.570765346 + 0.2122931212i\)
\(L(1)\) \(\approx\) \(1.570765346 + 0.2122931212i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.994 - 0.104i)T \)
3 \( 1 + (-0.406 + 0.913i)T \)
7 \( 1 + (-0.0523 + 0.998i)T \)
11 \( 1 + (-0.629 + 0.777i)T \)
13 \( 1 + (-0.0523 - 0.998i)T \)
17 \( 1 + (0.707 - 0.707i)T \)
19 \( 1 + (-0.544 - 0.838i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
29 \( 1 + (-0.866 - 0.5i)T \)
31 \( 1 + (-0.258 - 0.965i)T \)
37 \( 1 + (-0.965 + 0.258i)T \)
41 \( 1 + (0.587 + 0.809i)T \)
43 \( 1 + (-0.891 - 0.453i)T \)
47 \( 1 - iT \)
53 \( 1 + (0.866 - 0.5i)T \)
59 \( 1 + (0.866 + 0.5i)T \)
61 \( 1 + iT \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + (-0.258 - 0.965i)T \)
73 \( 1 + (0.891 - 0.453i)T \)
79 \( 1 + (0.951 - 0.309i)T \)
83 \( 1 + (0.913 - 0.406i)T \)
89 \( 1 + (0.777 + 0.629i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−17.50418800795308867725057587982, −16.97245172268294429198356736495, −16.38911403684486792443091078057, −16.0935754746791585219979811024, −14.75069393735287946474528085085, −14.372216567697213587227302901789, −13.775881820854093192425564821899, −13.16735535739910773077523639568, −12.57771713262583986198408715442, −12.11091012000705223101974840008, −11.128031133535645677251787137919, −10.798138790868938653845763226015, −10.21926941202381501373288129721, −8.851598358244838325975066512, −8.06662266251057726723120444648, −7.53392811028148256881806802397, −6.75814774770574167642536161018, −6.402911239988946382432293292682, −5.49721720803120716993805590508, −5.019068970378289967204106068974, −3.974783739818190934836329446622, −3.46894031307533747506961530706, −2.50560686068755522985149312479, −1.6826683093930933341554518729, −1.02887021345415672342535800505, 0.44571311354895039916349689406, 1.860033206125514325648979396427, 2.68796831527894220216126918541, 3.17619495817266062795152809901, 4.011202608083675226381486459169, 4.91991560070299651354161043017, 5.32078020472660454117066949253, 5.72741590703661907394224409033, 6.66765793127278717569460331455, 7.43817129185389198141813094001, 8.24382607244357091544170416931, 9.25746996462436927427448876821, 9.8054568219834686421256724630, 10.50436618044839218792674489430, 11.129489946537935514527121060483, 11.87829413181047126837814341794, 12.22011244046005052118870656563, 13.1640088272243216676286557108, 13.499592103549686114587519690101, 14.88111197969775088868555181358, 15.07459802173491964691555313752, 15.3131966022986020312619259812, 16.21989573393259926010723111460, 16.71820470349397836012206449139, 17.6243283454396297493163391761

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