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Properties

Label 1-6025-6025.194-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.811 - 0.583i$
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  + (0.587 + 0.809i)2-s − i·3-s + (−0.309 + 0.951i)4-s + (0.809 − 0.587i)6-s + (−0.987 + 0.156i)7-s + (−0.951 + 0.309i)8-s − 9-s + (0.156 − 0.987i)11-s + (0.951 + 0.309i)12-s + (0.891 − 0.453i)13-s + (−0.707 − 0.707i)14-s + (−0.809 − 0.587i)16-s + (0.891 − 0.453i)17-s + (−0.587 − 0.809i)18-s + (0.891 + 0.453i)19-s + ⋯
L(s)  = 1  + (0.587 + 0.809i)2-s − i·3-s + (−0.309 + 0.951i)4-s + (0.809 − 0.587i)6-s + (−0.987 + 0.156i)7-s + (−0.951 + 0.309i)8-s − 9-s + (0.156 − 0.987i)11-s + (0.951 + 0.309i)12-s + (0.891 − 0.453i)13-s + (−0.707 − 0.707i)14-s + (−0.809 − 0.587i)16-s + (0.891 − 0.453i)17-s + (−0.587 − 0.809i)18-s + (0.891 + 0.453i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.811 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.811 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.811 - 0.583i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (194, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.811 - 0.583i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.650534621 - 0.5318400525i\)
\(L(\frac12)\) \(\approx\) \(1.650534621 - 0.5318400525i\)
\(L(1)\) \(\approx\) \(1.194165208 + 0.09417783769i\)
\(L(1)\) \(\approx\) \(1.194165208 + 0.09417783769i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.587 + 0.809i)T \)
3 \( 1 - iT \)
7 \( 1 + (-0.987 + 0.156i)T \)
11 \( 1 + (0.156 - 0.987i)T \)
13 \( 1 + (0.891 - 0.453i)T \)
17 \( 1 + (0.891 - 0.453i)T \)
19 \( 1 + (0.891 + 0.453i)T \)
23 \( 1 + (-0.987 + 0.156i)T \)
29 \( 1 + (0.951 - 0.309i)T \)
31 \( 1 + (-0.891 + 0.453i)T \)
37 \( 1 + (-0.156 + 0.987i)T \)
41 \( 1 + (-0.951 + 0.309i)T \)
43 \( 1 + (0.987 + 0.156i)T \)
47 \( 1 + (-0.951 + 0.309i)T \)
53 \( 1 + (0.951 - 0.309i)T \)
59 \( 1 + (-0.587 - 0.809i)T \)
61 \( 1 + (-0.587 + 0.809i)T \)
67 \( 1 + (-0.951 - 0.309i)T \)
71 \( 1 + (0.453 - 0.891i)T \)
73 \( 1 + (0.891 + 0.453i)T \)
79 \( 1 + (0.587 - 0.809i)T \)
83 \( 1 + T \)
89 \( 1 + (0.987 + 0.156i)T \)
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

−17.965844281165319436567802227464, −16.975221085078024496290208252159, −16.264230871054649080296423413142, −15.73975316086077140044888551190, −15.17297292762505447007040217344, −14.34667977835585859242305528493, −13.92012113338240155035245635133, −13.18453034523089169343852756258, −12.32292682167310565366828088548, −11.95879601061120704378094292913, −11.114542420116022750424629624295, −10.34367677927645744607334938171, −10.063569057380449005140828378901, −9.25570866276494109433064872863, −8.93817071776465696844732470377, −7.75213588232129752662407631787, −6.725582255685538657934470904228, −6.04695837818968317282273781259, −5.41921758872455869004669927322, −4.648827254837094125855529202299, −3.7762026796958913972473484168, −3.63869792818896136423714083225, −2.70389395307870316072280822373, −1.88416192411927770930564751333, −0.812807255515396269480730019448, 0.458552234031994407100207955489, 1.39272732665795131206318026078, 2.66035175380797387222839354602, 3.39280663979541376315852103566, 3.55482636412412052509320978488, 5.01459215297661301969705119642, 5.702864982044624916893677307402, 6.194598090946813305089851372379, 6.630596211371292983108917658723, 7.60468233698879101540499323681, 8.02085889534927046580981063473, 8.72162682879815213567090836109, 9.40949190743142578698701063776, 10.3471483304677366016117329145, 11.3811762790860834819638576391, 12.05847408000917630902653425685, 12.3923648566137384353300825718, 13.331305297518312046088686824747, 13.670011274746764719934628846849, 14.116270778001063657698855747081, 14.969089702857441811102977388664, 15.84520843770421080877945371859, 16.36500337440002661532788418886, 16.68308954928896584468000998248, 17.74096737586167534669335432931

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