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Properties

Label 1-6025-6025.1347-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.729 - 0.684i$
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  + (0.358 − 0.933i)2-s + (0.933 − 0.358i)3-s + (−0.743 − 0.669i)4-s − i·6-s + (−0.688 + 0.725i)7-s + (−0.891 + 0.453i)8-s + (0.743 − 0.669i)9-s + (−0.725 − 0.688i)11-s + (−0.933 − 0.358i)12-s + (0.477 − 0.878i)13-s + (0.430 + 0.902i)14-s + (0.104 + 0.994i)16-s + (−0.972 + 0.233i)17-s + (−0.358 − 0.933i)18-s + (−0.902 − 0.430i)19-s + ⋯
L(s)  = 1  + (0.358 − 0.933i)2-s + (0.933 − 0.358i)3-s + (−0.743 − 0.669i)4-s − i·6-s + (−0.688 + 0.725i)7-s + (−0.891 + 0.453i)8-s + (0.743 − 0.669i)9-s + (−0.725 − 0.688i)11-s + (−0.933 − 0.358i)12-s + (0.477 − 0.878i)13-s + (0.430 + 0.902i)14-s + (0.104 + 0.994i)16-s + (−0.972 + 0.233i)17-s + (−0.358 − 0.933i)18-s + (−0.902 − 0.430i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.729 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.729 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.729 - 0.684i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (1347, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.729 - 0.684i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.710807683 - 0.6771643493i\)
\(L(\frac12)\) \(\approx\) \(1.710807683 - 0.6771643493i\)
\(L(1)\) \(\approx\) \(1.102828607 - 0.6799504007i\)
\(L(1)\) \(\approx\) \(1.102828607 - 0.6799504007i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.358 - 0.933i)T \)
3 \( 1 + (0.933 - 0.358i)T \)
7 \( 1 + (-0.688 + 0.725i)T \)
11 \( 1 + (-0.725 - 0.688i)T \)
13 \( 1 + (0.477 - 0.878i)T \)
17 \( 1 + (-0.972 + 0.233i)T \)
19 \( 1 + (-0.902 - 0.430i)T \)
23 \( 1 + (0.0784 + 0.996i)T \)
29 \( 1 + (-0.358 + 0.933i)T \)
31 \( 1 + (0.958 + 0.284i)T \)
37 \( 1 + (-0.983 + 0.182i)T \)
41 \( 1 + (0.707 + 0.707i)T \)
43 \( 1 + (0.233 + 0.972i)T \)
47 \( 1 + (0.156 + 0.987i)T \)
53 \( 1 + (0.629 + 0.777i)T \)
59 \( 1 + (0.544 + 0.838i)T \)
61 \( 1 + (-0.453 - 0.891i)T \)
67 \( 1 + (0.777 + 0.629i)T \)
71 \( 1 + (-0.0261 - 0.999i)T \)
73 \( 1 + (0.522 - 0.852i)T \)
79 \( 1 + (0.707 + 0.707i)T \)
83 \( 1 + (-0.913 + 0.406i)T \)
89 \( 1 + (-0.0261 - 0.999i)T \)
97 \( 1 + (0.104 - 0.994i)T \)
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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

−17.54197619047024423627304146125, −16.99074367772984881686292758436, −16.237930357841156580378336135127, −15.78095309361501819323319733506, −15.22823121116793449361769299675, −14.58275252613359512850028869187, −13.759353492102390907502361234834, −13.531246796466305835446441007921, −12.82135524549934662417563713227, −12.22542003302801052976770016516, −11.022559135945181139925488990, −10.23142647011594178099612105055, −9.73812138270508304847995009284, −8.88716096337057488428298720340, −8.4585087525915065532356923478, −7.6898137228626953542273361060, −6.86383647226637335566749470919, −6.66915041203208706405779018070, −5.56589935660656882967721674079, −4.61280118722110055376064166563, −4.11974591471706003713377396878, −3.70593223396482829074628290557, −2.57544540586530043609530377101, −2.064925477962073515242497934418, −0.43129012270900009752420886763, 0.798197722609390965647939052, 1.71042224911118320427216597075, 2.58932799982990952834353682248, 2.96708779125166904972314253311, 3.58495823231241967849011733543, 4.45134755846316807217647143452, 5.346438170040945300622485345055, 6.08451612976438855879870146648, 6.67457069046269605853297561214, 7.84905355990312453483792054575, 8.488044733222641452671718127156, 9.003566158103760964476134941460, 9.57043720285825859871368544466, 10.485654531236796201991718865185, 10.91552372143665688732690122410, 11.81863453232890884783669329652, 12.65123090429482945922570738203, 13.036666035497576667702689896664, 13.423387603366872836947008375518, 14.15375455845679739401275770095, 14.99952168736102957149504006808, 15.55108153248224966252887419128, 15.87210562093471589702838519713, 17.24495927806264102764876910310, 18.06444063451897742084515831294

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