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Properties

Label 1-6025-6025.138-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.795 - 0.605i$
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  + (−0.891 − 0.453i)2-s + (0.987 − 0.156i)3-s + (0.587 + 0.809i)4-s + (−0.951 − 0.309i)6-s + (−0.972 − 0.233i)7-s + (−0.156 − 0.987i)8-s + (0.951 − 0.309i)9-s + (0.760 − 0.649i)11-s + (0.707 + 0.707i)12-s + (0.923 + 0.382i)13-s + (0.760 + 0.649i)14-s + (−0.309 + 0.951i)16-s + (0.0784 − 0.996i)17-s + (−0.987 − 0.156i)18-s + (0.972 + 0.233i)19-s + ⋯
L(s)  = 1  + (−0.891 − 0.453i)2-s + (0.987 − 0.156i)3-s + (0.587 + 0.809i)4-s + (−0.951 − 0.309i)6-s + (−0.972 − 0.233i)7-s + (−0.156 − 0.987i)8-s + (0.951 − 0.309i)9-s + (0.760 − 0.649i)11-s + (0.707 + 0.707i)12-s + (0.923 + 0.382i)13-s + (0.760 + 0.649i)14-s + (−0.309 + 0.951i)16-s + (0.0784 − 0.996i)17-s + (−0.987 − 0.156i)18-s + (0.972 + 0.233i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.795 - 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.795 - 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.795 - 0.605i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (138, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.795 - 0.605i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.822823613 - 0.6143540389i\)
\(L(\frac12)\) \(\approx\) \(1.822823613 - 0.6143540389i\)
\(L(1)\) \(\approx\) \(1.054483692 - 0.2727816834i\)
\(L(1)\) \(\approx\) \(1.054483692 - 0.2727816834i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.891 - 0.453i)T \)
3 \( 1 + (0.987 - 0.156i)T \)
7 \( 1 + (-0.972 - 0.233i)T \)
11 \( 1 + (0.760 - 0.649i)T \)
13 \( 1 + (0.923 + 0.382i)T \)
17 \( 1 + (0.0784 - 0.996i)T \)
19 \( 1 + (0.972 + 0.233i)T \)
23 \( 1 + (-0.522 + 0.852i)T \)
29 \( 1 + (0.453 + 0.891i)T \)
31 \( 1 + (-0.996 - 0.0784i)T \)
37 \( 1 + (0.972 + 0.233i)T \)
41 \( 1 + (-0.987 - 0.156i)T \)
43 \( 1 + (-0.233 - 0.972i)T \)
47 \( 1 + (0.891 - 0.453i)T \)
53 \( 1 + (0.453 + 0.891i)T \)
59 \( 1 + (0.156 + 0.987i)T \)
61 \( 1 + (-0.156 + 0.987i)T \)
67 \( 1 + (0.891 + 0.453i)T \)
71 \( 1 + (-0.760 - 0.649i)T \)
73 \( 1 + (0.923 - 0.382i)T \)
79 \( 1 + (-0.453 + 0.891i)T \)
83 \( 1 + (-0.809 + 0.587i)T \)
89 \( 1 + (0.996 + 0.0784i)T \)
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

−17.886216518119427497492259105659, −17.0666529566295427895891710338, −16.336834076753997563162570052179, −15.82182404890669270381585463383, −15.320402829679888527345056484010, −14.621818093396589098319386568393, −14.11430734249664374535130704446, −13.21005257262077651901174282788, −12.66066792730446619375560678576, −11.77738095599827070873925183684, −10.91049160913625595048626691023, −10.079887160436663284545209210236, −9.74451588384488706638226749166, −9.09051961043338137199708705376, −8.46903845084964030952202221020, −7.894836123701922842399458012877, −7.11791452153550145343165696848, −6.41355279239001234808832652550, −5.94003151376812450431134329787, −4.83776518820173467332938134803, −3.88436124149267194870064449095, −3.26609106135106686530322788153, −2.36082529575518420674267082489, −1.67703918363005780128303962726, −0.766414632895657013589986133491, 0.83972651702052577108813744228, 1.36180838781973486158486943230, 2.33725062522133603761332869449, 3.20780676585452264051278481866, 3.54084271962561622039735133942, 4.16449678790738001000129637728, 5.630545371496127268483955926340, 6.45744040097703672613135454206, 7.18536608977319078824001864123, 7.51722947779560334574588476933, 8.68343120813334416470329998928, 8.87674566431271581647448233640, 9.596561331941800193260025193226, 10.08056172008951736174140560831, 10.92641323183483339284173501740, 11.76335750948679807743964989439, 12.19804358843063780646891646201, 13.15641600804874610570636527401, 13.68964573832942606859887391700, 14.08874537259492987336259940869, 15.24604337151031497291089956729, 15.8529118563461841942481853527, 16.39798977972417642267132828436, 16.81605838698357988720068886286, 17.99765027571876474426588041131

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