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Properties

Label 1-6025-6025.341-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.662 + 0.749i$
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  + (−0.978 − 0.207i)2-s + (0.669 − 0.743i)3-s + (0.913 + 0.406i)4-s + (−0.809 + 0.587i)6-s + (−0.104 − 0.994i)7-s + (−0.809 − 0.587i)8-s + (−0.104 − 0.994i)9-s + (−0.978 − 0.207i)11-s + (0.913 − 0.406i)12-s + (−0.104 + 0.994i)13-s + (−0.104 + 0.994i)14-s + (0.669 + 0.743i)16-s + 17-s + (−0.104 + 0.994i)18-s + (0.913 − 0.406i)19-s + ⋯
L(s)  = 1  + (−0.978 − 0.207i)2-s + (0.669 − 0.743i)3-s + (0.913 + 0.406i)4-s + (−0.809 + 0.587i)6-s + (−0.104 − 0.994i)7-s + (−0.809 − 0.587i)8-s + (−0.104 − 0.994i)9-s + (−0.978 − 0.207i)11-s + (0.913 − 0.406i)12-s + (−0.104 + 0.994i)13-s + (−0.104 + 0.994i)14-s + (0.669 + 0.743i)16-s + 17-s + (−0.104 + 0.994i)18-s + (0.913 − 0.406i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.662 + 0.749i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.662 + 0.749i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6139980633 + 0.2768649955i\)
\(L(\frac12)\) \(\approx\) \(0.6139980633 + 0.2768649955i\)
\(L(1)\) \(\approx\) \(0.7208865583 - 0.2431169082i\)
\(L(1)\) \(\approx\) \(0.7208865583 - 0.2431169082i\)

Euler product

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

−17.66315724054202945776637347521, −16.88814959767984530682314182475, −16.22351858494955940281041235093, −15.62964443193119654582133862830, −15.205676814926964545619127346162, −14.7286019258742713826641607616, −13.89363679559967452247933162631, −12.93943742179633659625999953875, −12.314436507627297271607709722355, −11.456304588644055879111596358995, −10.717374690567003077776755133131, −10.18385264607032405317657977182, −9.50323075905964737624361758545, −9.11443669546377058729254055396, −8.22695313931391973856737243320, −7.75207256278122133366659668394, −7.29855708356075427746449187066, −5.858425064760002371972465248776, −5.55763095706022096967529572169, −4.92226730872630188978985567142, −3.56341618752541391810996633514, −2.936988096681785635061348880247, −2.41217772340419755489135953751, −1.54586325912648991912474441135, −0.217130353863303273083555651717, 1.10302405083607767415972974108, 1.368873747276886058216604997044, 2.4749018312157296912592858492, 3.18014397746349811327659372203, 3.62843574729864016946722549068, 4.86496225945942098321292606429, 5.81170287781860751631035116846, 6.87835900297329712701493490782, 7.16719935638333571927518730236, 7.654650147514212886678121926862, 8.52705924489989432271434360301, 8.98851335708938735282891272415, 9.87443165586204702860966564560, 10.27045047917914884045051768024, 11.243686578604579356608635913649, 11.68518552627064828390412960882, 12.635049711159059704827906829846, 13.07845939534180377453115532532, 13.8658634279944966962156831988, 14.44526271862816372927404924322, 15.22420096356841600679815478800, 16.08261997460217436561117140726, 16.58558169604638931657791412420, 17.21982715356649774592339679011, 17.99693583484355195611676927182

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