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Properties

Label 1-6025-6025.592-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.718 + 0.695i$
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  + (0.777 + 0.629i)2-s + (0.0523 + 0.998i)3-s + (0.207 + 0.978i)4-s + (−0.587 + 0.809i)6-s + (−0.182 − 0.983i)7-s + (−0.453 + 0.891i)8-s + (−0.994 + 0.104i)9-s + (0.477 + 0.878i)11-s + (−0.965 + 0.258i)12-s + (−0.991 + 0.130i)13-s + (0.477 − 0.878i)14-s + (−0.913 + 0.406i)16-s + (−0.233 − 0.972i)17-s + (−0.838 − 0.544i)18-s + (−0.182 − 0.983i)19-s + ⋯
L(s)  = 1  + (0.777 + 0.629i)2-s + (0.0523 + 0.998i)3-s + (0.207 + 0.978i)4-s + (−0.587 + 0.809i)6-s + (−0.182 − 0.983i)7-s + (−0.453 + 0.891i)8-s + (−0.994 + 0.104i)9-s + (0.477 + 0.878i)11-s + (−0.965 + 0.258i)12-s + (−0.991 + 0.130i)13-s + (0.477 − 0.878i)14-s + (−0.913 + 0.406i)16-s + (−0.233 − 0.972i)17-s + (−0.838 − 0.544i)18-s + (−0.182 − 0.983i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.718 + 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.718 + 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.718 + 0.695i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (592, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.718 + 0.695i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.841952262 + 0.7457234453i\)
\(L(\frac12)\) \(\approx\) \(1.841952262 + 0.7457234453i\)
\(L(1)\) \(\approx\) \(1.163195025 + 0.7647082716i\)
\(L(1)\) \(\approx\) \(1.163195025 + 0.7647082716i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.777 + 0.629i)T \)
3 \( 1 + (0.0523 + 0.998i)T \)
7 \( 1 + (-0.182 - 0.983i)T \)
11 \( 1 + (0.477 + 0.878i)T \)
13 \( 1 + (-0.991 + 0.130i)T \)
17 \( 1 + (-0.233 - 0.972i)T \)
19 \( 1 + (-0.182 - 0.983i)T \)
23 \( 1 + (0.996 - 0.0784i)T \)
29 \( 1 + (0.358 - 0.933i)T \)
31 \( 1 + (-0.284 + 0.958i)T \)
37 \( 1 + (0.182 + 0.983i)T \)
41 \( 1 + (-0.891 + 0.453i)T \)
43 \( 1 + (0.649 - 0.760i)T \)
47 \( 1 + (-0.156 - 0.987i)T \)
53 \( 1 + (-0.629 - 0.777i)T \)
59 \( 1 + (-0.544 - 0.838i)T \)
61 \( 1 + (0.453 + 0.891i)T \)
67 \( 1 + (-0.777 - 0.629i)T \)
71 \( 1 + (0.999 - 0.0261i)T \)
73 \( 1 + (0.382 + 0.923i)T \)
79 \( 1 + (0.987 + 0.156i)T \)
83 \( 1 + (-0.669 - 0.743i)T \)
89 \( 1 + (0.284 - 0.958i)T \)
97 \( 1 + (0.104 - 0.994i)T \)
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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

−17.81604070905576873981789502626, −16.99895654437411724158216635668, −16.3480282498861956484145961744, −15.35576763688485051285630899172, −14.71840899365282406324981584770, −14.37461599662690476768218882548, −13.57993473927526091185010776049, −12.789905093571487243205160943584, −12.469861677477801948189849052, −11.97323662117654466517556540412, −11.12839533432001541143769872556, −10.72104174460233304473171734855, −9.56057174049765818522374907479, −9.05222903325949440571555068650, −8.2926254958955885633717043019, −7.463615558853428716693053098030, −6.562047705246047475234078659247, −6.02776940444239931096734993219, −5.55908891470965464308450526572, −4.73022206663994367291134967840, −3.64790775341795410346928132366, −3.05870939558134658011340676579, −2.32692751088380950823665649085, −1.70605078125914620613491380538, −0.85666967442957352213169925725, 0.4213526562018100238180556872, 2.04014081068354660874764344181, 2.89425033319245526955062997260, 3.467264301561585677486710537172, 4.41395700671131659590062377065, 4.75215692460001130995426277236, 5.19602953255195888304889918116, 6.44273170123689983477170692846, 6.94561530965082275283374460648, 7.46050234664774728365132407843, 8.42445340108733584064354260782, 9.17753686736375503729983560929, 9.77686783965261086456595468115, 10.45632935695883052163673334229, 11.38684738212423117674685123119, 11.75833905480878882122614347873, 12.6771929194156928328360247618, 13.37961787361962406488146016322, 14.06431997632039281932691778203, 14.49961260646117391294111483592, 15.33140701399567734543021141062, 15.55557550088988339683739710776, 16.500370067022417055786610708, 17.054800504834641970611850406241, 17.29587205887374016421732683278

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