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Properties

Label 1-6025-6025.359-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.0193 - 0.999i$
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 + (−0.978 + 0.207i)3-s + (0.913 + 0.406i)4-s + 6-s + (0.207 − 0.978i)7-s + (−0.809 − 0.587i)8-s + (0.913 − 0.406i)9-s + (−0.207 + 0.978i)11-s + (−0.978 − 0.207i)12-s + (−0.743 − 0.669i)13-s + (−0.406 + 0.913i)14-s + (0.669 + 0.743i)16-s + (−0.951 − 0.309i)17-s + (−0.978 + 0.207i)18-s + (−0.406 − 0.913i)19-s + ⋯
L(s)  = 1  + (−0.978 − 0.207i)2-s + (−0.978 + 0.207i)3-s + (0.913 + 0.406i)4-s + 6-s + (0.207 − 0.978i)7-s + (−0.809 − 0.587i)8-s + (0.913 − 0.406i)9-s + (−0.207 + 0.978i)11-s + (−0.978 − 0.207i)12-s + (−0.743 − 0.669i)13-s + (−0.406 + 0.913i)14-s + (0.669 + 0.743i)16-s + (−0.951 − 0.309i)17-s + (−0.978 + 0.207i)18-s + (−0.406 − 0.913i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0193 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0193 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.0193 - 0.999i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (359, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.0193 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3844179095 - 0.3770489317i\)
\(L(\frac12)\) \(\approx\) \(0.3844179095 - 0.3770489317i\)
\(L(1)\) \(\approx\) \(0.4804336592 - 0.09464075183i\)
\(L(1)\) \(\approx\) \(0.4804336592 - 0.09464075183i\)

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 \( 1 + (-0.978 + 0.207i)T \)
7 \( 1 + (0.207 - 0.978i)T \)
11 \( 1 + (-0.207 + 0.978i)T \)
13 \( 1 + (-0.743 - 0.669i)T \)
17 \( 1 + (-0.951 - 0.309i)T \)
19 \( 1 + (-0.406 - 0.913i)T \)
23 \( 1 + (0.587 - 0.809i)T \)
29 \( 1 + (0.978 + 0.207i)T \)
31 \( 1 + (-0.743 + 0.669i)T \)
37 \( 1 + (0.994 - 0.104i)T \)
41 \( 1 - T \)
43 \( 1 + (-0.951 - 0.309i)T \)
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 \( 1 + (0.207 - 0.978i)T \)
73 \( 1 + (0.951 + 0.309i)T \)
79 \( 1 - T \)
83 \( 1 + (0.978 + 0.207i)T \)
89 \( 1 + (0.207 - 0.978i)T \)
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

−17.91428642734236875607210502158, −17.12785849125648232585114340677, −16.752217578321782206282305167, −16.12769325987310350567546521045, −15.380203462925533236494700113924, −14.97508556834768625877684741972, −14.007519247017300852493848370954, −13.1081569598867225388536892403, −12.34948411794144818512267896420, −11.67659625756553696260211388024, −11.306241970548912418421042943335, −10.66662603484550250089794850232, −9.82976626639092776452998602976, −9.29623562479284246326146407694, −8.369370470680595135023285775958, −8.031474028308447608951350163936, −6.94025993763212204145128116334, −6.53846192880278342486947387225, −5.734365677904656649223054544635, −5.32971989690606825176569544085, −4.40113300807019608106104203990, −3.23530365241391934998477666160, −2.16498116501026477140369305419, −1.791543036565522123692450630199, −0.665019646032002160545787380412, 0.36133062869567479221169793820, 1.04643061971560539071123277994, 2.067257377099996896213453857690, 2.804421304949660867904248034382, 3.89845784317949805064317144643, 4.70146128745325763322139398768, 5.13521407265383761428908451328, 6.39425793369581625615737253089, 6.98312607288996674818606166693, 7.23000383256699207039740777201, 8.18013299570140398851197669839, 9.01473659878143850907230836060, 9.779378272394544928595294696207, 10.39199081680471288061786549499, 10.69757369909125002123923794592, 11.43930172773225546420028936115, 12.07498077378848048743476057659, 12.874416253311741289482940682068, 13.19975109784587631503923258291, 14.57355779641547079527922674321, 15.164584486529391725418655400568, 15.75730487903758427648157073842, 16.509243786645955150589266207717, 17.02077453306539435238199105544, 17.71387420973914198201516654286

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