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Properties

Label 1-6025-6025.979-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.539 - 0.841i$
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  + (−0.913 + 0.406i)2-s + (−0.669 + 0.743i)3-s + (0.669 − 0.743i)4-s + (0.309 − 0.951i)6-s + (0.5 − 0.866i)7-s + (−0.309 + 0.951i)8-s + (−0.104 − 0.994i)9-s + (0.913 − 0.406i)11-s + (0.104 + 0.994i)12-s + (−0.913 − 0.406i)13-s + (−0.104 + 0.994i)14-s + (−0.104 − 0.994i)16-s + (−0.309 + 0.951i)17-s + (0.5 + 0.866i)18-s + (0.669 + 0.743i)19-s + ⋯
L(s)  = 1  + (−0.913 + 0.406i)2-s + (−0.669 + 0.743i)3-s + (0.669 − 0.743i)4-s + (0.309 − 0.951i)6-s + (0.5 − 0.866i)7-s + (−0.309 + 0.951i)8-s + (−0.104 − 0.994i)9-s + (0.913 − 0.406i)11-s + (0.104 + 0.994i)12-s + (−0.913 − 0.406i)13-s + (−0.104 + 0.994i)14-s + (−0.104 − 0.994i)16-s + (−0.309 + 0.951i)17-s + (0.5 + 0.866i)18-s + (0.669 + 0.743i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.539 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.539 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.539 - 0.841i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (979, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.539 - 0.841i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05174851401 - 0.09464993288i\)
\(L(\frac12)\) \(\approx\) \(0.05174851401 - 0.09464993288i\)
\(L(1)\) \(\approx\) \(0.5250197396 + 0.1369164454i\)
\(L(1)\) \(\approx\) \(0.5250197396 + 0.1369164454i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.913 + 0.406i)T \)
3 \( 1 + (-0.669 + 0.743i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (0.913 - 0.406i)T \)
13 \( 1 + (-0.913 - 0.406i)T \)
17 \( 1 + (-0.309 + 0.951i)T \)
19 \( 1 + (0.669 + 0.743i)T \)
23 \( 1 + (0.809 + 0.587i)T \)
29 \( 1 + (-0.978 - 0.207i)T \)
31 \( 1 + (0.669 + 0.743i)T \)
37 \( 1 + (0.104 + 0.994i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.309 - 0.951i)T \)
53 \( 1 + (-0.669 + 0.743i)T \)
59 \( 1 + (-0.104 - 0.994i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + (-0.669 - 0.743i)T \)
71 \( 1 + (-0.978 - 0.207i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (-0.669 - 0.743i)T \)
89 \( 1 + (0.913 - 0.406i)T \)
97 \( 1 + (-0.669 + 0.743i)T \)
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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

−17.83934899591523175151140720445, −17.56407683209218243776645494175, −16.75387409386229507051759230807, −16.36056875184739175262371773356, −15.38725165054862016586617448894, −14.80037302749527874579648576957, −13.90216937100812334741429440572, −13.0555086534052695907508326519, −12.40148594102412483155337192992, −11.6953056698399265869484876452, −11.59898884362769606607585736216, −10.84370372861098797528982130803, −9.87491725099459512196246087789, −9.193006261229482528198818229411, −8.79185746340071161872046938937, −7.76503440145609635181584866746, −7.2279635346914096902842528819, −6.74408847578219014175633010263, −5.92373446506668623940365995157, −4.988985709200940522612141982623, −4.439774415667694545437356296236, −3.078371892020480724788879991002, −2.38749224902056117296490214554, −1.80941005423272110602491742765, −1.00580216696329282828794228488, 0.04688676040053488371914136489, 1.19655899226011764326807554987, 1.6347440654680903317027529511, 3.13436539969611930360901333166, 3.7188907331223265160868905958, 4.81649738577335736005406377002, 5.17286829547018424027435769755, 6.18817348381289043491650766938, 6.6247241025008278547558269477, 7.46102503135681135932764452311, 8.119159921680774171338447306107, 8.88825915473534837206803885492, 9.625901375563858797942537864091, 10.17531854405137621349885026494, 10.64412860239563201507683175479, 11.55526415229852777299226267639, 11.68119267395836829409355145850, 12.761415175610725992058762391124, 13.8370223790997164844198100378, 14.46876324306959384645173552219, 15.12294610133920201009558375962, 15.479927767631743549465068957959, 16.62240222323587066949693269007, 16.86501504091828436662747578196, 17.21796014430138739624719562943

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