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Properties

Label 1-6025-6025.4756-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.874 + 0.485i$
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  + (0.809 − 0.587i)2-s + (−0.309 + 0.951i)3-s + (0.309 − 0.951i)4-s + (0.309 + 0.951i)6-s + i·7-s + (−0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + (−0.587 − 0.809i)11-s + (0.809 + 0.587i)12-s + (−0.587 + 0.809i)13-s + (0.587 + 0.809i)14-s + (−0.809 − 0.587i)16-s + (0.951 − 0.309i)17-s − 18-s + (−0.951 + 0.309i)19-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)2-s + (−0.309 + 0.951i)3-s + (0.309 − 0.951i)4-s + (0.309 + 0.951i)6-s + i·7-s + (−0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + (−0.587 − 0.809i)11-s + (0.809 + 0.587i)12-s + (−0.587 + 0.809i)13-s + (0.587 + 0.809i)14-s + (−0.809 − 0.587i)16-s + (0.951 − 0.309i)17-s − 18-s + (−0.951 + 0.309i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.874 + 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.874 + 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.874 + 0.485i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (4756, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.874 + 0.485i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.805529878 + 0.4680465039i\)
\(L(\frac12)\) \(\approx\) \(1.805529878 + 0.4680465039i\)
\(L(1)\) \(\approx\) \(1.284176594 - 0.03265137348i\)
\(L(1)\) \(\approx\) \(1.284176594 - 0.03265137348i\)

Euler product

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

−17.48066365942043476509693078858, −17.193080163397652510347723934964, −16.35366859677743223410337509092, −15.70780740680173419455915022711, −14.711058061375210478648414574, −14.54709118296761932436021452259, −13.48464707144270217658683303645, −13.11820193002229605045356820069, −12.69948607949319328175512530085, −11.848124613459476851739343996821, −11.39805331865960408980736779931, −10.32869304189738677498153189619, −9.960628930299007287238720397544, −8.53909869654224284822449727324, −7.82214303919862830552241199560, −7.601645953509073577281819445925, −6.88206204692206383048899496781, −6.18241840688813801304417979208, −5.51219983140766657433147176006, −4.80499334419647448272952270703, −4.12622107910406266063330170194, −3.16829974980470487086974496028, −2.45522965116839184216039303153, −1.65322015871353347535684156811, −0.47968253811980465216958467606, 0.71513509733090395401003155488, 2.01737554584437170784371700827, 2.67958666032116077975712939622, 3.27041361381371358141344562003, 4.18204891647862176275506478521, 4.71826893791548797381049852100, 5.47803757655374539283942753476, 5.965962144046870425275554468690, 6.53840215197448809079644796047, 7.76833070588671996752163859657, 8.69595215527800096693256936979, 9.27855176671340541101785696865, 9.96186868372118732448426106943, 10.593726177900141737087920751811, 11.20087333730238504350142984453, 11.86590142427794945583788702882, 12.35893891517506181837540718501, 12.97890671574466373428941762567, 14.173889302966864455003336665012, 14.34827682440003417180441810734, 15.07050416719256945215244140558, 15.7204950380186220796516442991, 16.36490143111506993169445411329, 16.734519583345343721294965926533, 17.91716555807166027433197019495

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