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Properties

Label 1-6025-6025.1222-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.212 - 0.977i$
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  + (−0.156 − 0.987i)2-s + (0.707 + 0.707i)3-s + (−0.951 + 0.309i)4-s + (0.587 − 0.809i)6-s + (−0.233 − 0.972i)7-s + (0.453 + 0.891i)8-s + i·9-s + (0.522 + 0.852i)11-s + (−0.891 − 0.453i)12-s + (0.760 − 0.649i)13-s + (−0.923 + 0.382i)14-s + (0.809 − 0.587i)16-s + (−0.760 + 0.649i)17-s + (0.987 − 0.156i)18-s + (−0.760 − 0.649i)19-s + ⋯
L(s)  = 1  + (−0.156 − 0.987i)2-s + (0.707 + 0.707i)3-s + (−0.951 + 0.309i)4-s + (0.587 − 0.809i)6-s + (−0.233 − 0.972i)7-s + (0.453 + 0.891i)8-s + i·9-s + (0.522 + 0.852i)11-s + (−0.891 − 0.453i)12-s + (0.760 − 0.649i)13-s + (−0.923 + 0.382i)14-s + (0.809 − 0.587i)16-s + (−0.760 + 0.649i)17-s + (0.987 − 0.156i)18-s + (−0.760 − 0.649i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.212 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.212 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.212 - 0.977i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (1222, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.212 - 0.977i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.350101381 - 1.087705484i\)
\(L(\frac12)\) \(\approx\) \(1.350101381 - 1.087705484i\)
\(L(1)\) \(\approx\) \(1.056568365 - 0.3563664620i\)
\(L(1)\) \(\approx\) \(1.056568365 - 0.3563664620i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.156 - 0.987i)T \)
3 \( 1 + (0.707 + 0.707i)T \)
7 \( 1 + (-0.233 - 0.972i)T \)
11 \( 1 + (0.522 + 0.852i)T \)
13 \( 1 + (0.760 - 0.649i)T \)
17 \( 1 + (-0.760 + 0.649i)T \)
19 \( 1 + (-0.760 - 0.649i)T \)
23 \( 1 + (-0.233 - 0.972i)T \)
29 \( 1 + (0.891 - 0.453i)T \)
31 \( 1 + (-0.649 - 0.760i)T \)
37 \( 1 + (-0.852 + 0.522i)T \)
41 \( 1 + (0.453 + 0.891i)T \)
43 \( 1 + (0.972 + 0.233i)T \)
47 \( 1 + (0.453 + 0.891i)T \)
53 \( 1 + (0.891 - 0.453i)T \)
59 \( 1 + (-0.987 + 0.156i)T \)
61 \( 1 + (0.987 + 0.156i)T \)
67 \( 1 + (0.453 - 0.891i)T \)
71 \( 1 + (-0.0784 - 0.996i)T \)
73 \( 1 + (0.760 + 0.649i)T \)
79 \( 1 + (0.156 - 0.987i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.972 - 0.233i)T \)
97 \( 1 + (0.309 - 0.951i)T \)
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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

−17.950476432559350258073190779393, −17.28122230383901010392340006987, −16.366932048418112925710077702997, −15.807328919271517449670511658233, −15.363670581653275588050176670875, −14.44372766049623876181642887357, −14.00989435197715367380861382471, −13.563812841439950155598531952618, −12.702802461597568580324220130648, −12.17300423182628421250139883518, −11.348937752203879996295060700827, −10.42442804303176101467386850466, −9.36153122803469485377595738235, −8.83605577997662419857114732434, −8.709796157296617132442049779485, −7.847480625960263759729433804164, −6.93309014730214678347305045673, −6.56258869260206285109085328766, −5.83397154546771723128469396391, −5.26256705488397889341369438215, −3.935869927726512700055853751204, −3.6552025178550715755160643582, −2.51889797899575991719443014019, −1.711704681521496950703419643581, −0.82855375356452331123664264700, 0.52455671652186802965330686959, 1.58450366559693427555923420561, 2.32564695096814278036441321690, 3.055476116446783394905614883803, 3.90813655984927689726911457780, 4.30561711589531849850685284821, 4.78376676555803773501447282509, 6.02446441277767695000374148900, 6.88904924333033679638645554089, 7.84824017055719557887532722860, 8.36450049295420942826963884197, 9.075612621655884136312800170218, 9.66622612049532527583230801121, 10.40664970934566314489726988034, 10.73639859335779435004860535255, 11.33446785734563459268233873861, 12.47288280674267892965501556169, 12.96113426730187255044188500535, 13.567583336229656189372931893209, 14.16078225348965119825901493276, 14.887761304962792710883530967511, 15.47318622532536387206092776674, 16.35407268516832968853008561113, 17.06216245717471609468339624182, 17.55581569624540243908918730022

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