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Properties

Label 1-6025-6025.1233-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.999 + 0.0338i$
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 + (0.987 − 0.156i)3-s + (−0.951 − 0.309i)4-s − i·6-s + (−0.522 + 0.852i)7-s + (−0.453 + 0.891i)8-s + (0.951 − 0.309i)9-s + (0.852 + 0.522i)11-s + (−0.987 − 0.156i)12-s + (−0.972 + 0.233i)13-s + (0.760 + 0.649i)14-s + (0.809 + 0.587i)16-s + (0.522 + 0.852i)17-s + (−0.156 − 0.987i)18-s + (−0.649 − 0.760i)19-s + ⋯
L(s)  = 1  + (0.156 − 0.987i)2-s + (0.987 − 0.156i)3-s + (−0.951 − 0.309i)4-s − i·6-s + (−0.522 + 0.852i)7-s + (−0.453 + 0.891i)8-s + (0.951 − 0.309i)9-s + (0.852 + 0.522i)11-s + (−0.987 − 0.156i)12-s + (−0.972 + 0.233i)13-s + (0.760 + 0.649i)14-s + (0.809 + 0.587i)16-s + (0.522 + 0.852i)17-s + (−0.156 − 0.987i)18-s + (−0.649 − 0.760i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0338i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0338i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.999 + 0.0338i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (1233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.999 + 0.0338i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.283107274 + 0.03868754873i\)
\(L(\frac12)\) \(\approx\) \(2.283107274 + 0.03868754873i\)
\(L(1)\) \(\approx\) \(1.332441703 - 0.4282980509i\)
\(L(1)\) \(\approx\) \(1.332441703 - 0.4282980509i\)

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 \( 1 + (0.987 - 0.156i)T \)
7 \( 1 + (-0.522 + 0.852i)T \)
11 \( 1 + (0.852 + 0.522i)T \)
13 \( 1 + (-0.972 + 0.233i)T \)
17 \( 1 + (0.522 + 0.852i)T \)
19 \( 1 + (-0.649 - 0.760i)T \)
23 \( 1 + (0.649 + 0.760i)T \)
29 \( 1 + (0.156 - 0.987i)T \)
31 \( 1 + (0.852 - 0.522i)T \)
37 \( 1 + (0.0784 + 0.996i)T \)
41 \( 1 + (0.707 + 0.707i)T \)
43 \( 1 + (0.852 - 0.522i)T \)
47 \( 1 + (-0.987 - 0.156i)T \)
53 \( 1 + (0.156 - 0.987i)T \)
59 \( 1 + (-0.891 + 0.453i)T \)
61 \( 1 + (0.891 + 0.453i)T \)
67 \( 1 + (-0.987 + 0.156i)T \)
71 \( 1 + (0.233 + 0.972i)T \)
73 \( 1 + (-0.972 - 0.233i)T \)
79 \( 1 + (0.707 + 0.707i)T \)
83 \( 1 + (-0.809 + 0.587i)T \)
89 \( 1 + (0.233 + 0.972i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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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

−17.47911727099058421111234166078, −16.87976227912012205126549599048, −16.242286855415330983059249821795, −15.92665728848916178612699517461, −14.82420395793434845771970166100, −14.44201016211494346491471456617, −14.07830648029504569137167632832, −13.362366690617933838165679383270, −12.573537619076162342973583301967, −12.23405036121711288451347343915, −10.80003479730731997284929891641, −10.21643386127182452596353957149, −9.40990661250285305297661074580, −9.0861602949394236391035694859, −8.21315265454712392456041544846, −7.59201542900345924636039744736, −7.01021664827515474407774436061, −6.46326790000692386639221817208, −5.51950278113311689270243579908, −4.53456233854826140570360842346, −4.20143313593007788935263345270, −3.222039771388302191930910962979, −2.88238779286647495580221275562, −1.455919788751763742494147545111, −0.52175997817626880899380911207, 1.026523598430288848573802945896, 1.8276152102039073749983806328, 2.54913119653088604697776049012, 2.954971674657317438752987025807, 3.95008085805057980237873659585, 4.3970800908691300189433895360, 5.294055306995096597333781325702, 6.25979272468362291859311025806, 6.91301628026097116249412001025, 7.948520788792349922963677438241, 8.5214285298277756938239465537, 9.3097092365973486149030732738, 9.64896968275396767536479172310, 10.1608372416881078391361258897, 11.26389434434096788223030068473, 11.98635563344625585473883621089, 12.39543984529184393290996014704, 13.11653242201675090707343647335, 13.55630078716189296170290601439, 14.48617906455061224936991696006, 15.10853440073143254994687151702, 15.16184720878061316206351757786, 16.44320965526878930070390110219, 17.4281098577061497142796177398, 17.67802336356940486793830625037

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