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Properties

Label 1-6025-6025.739-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.987 - 0.157i$
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 + (−0.913 + 0.406i)3-s + (0.913 − 0.406i)4-s + (−0.809 + 0.587i)6-s + (−0.5 − 0.866i)7-s + (0.809 − 0.587i)8-s + (0.669 − 0.743i)9-s + (0.978 − 0.207i)11-s + (−0.669 + 0.743i)12-s + (−0.978 − 0.207i)13-s + (−0.669 − 0.743i)14-s + (0.669 − 0.743i)16-s + (−0.809 + 0.587i)17-s + (0.5 − 0.866i)18-s + (−0.913 − 0.406i)19-s + ⋯
L(s)  = 1  + (0.978 − 0.207i)2-s + (−0.913 + 0.406i)3-s + (0.913 − 0.406i)4-s + (−0.809 + 0.587i)6-s + (−0.5 − 0.866i)7-s + (0.809 − 0.587i)8-s + (0.669 − 0.743i)9-s + (0.978 − 0.207i)11-s + (−0.669 + 0.743i)12-s + (−0.978 − 0.207i)13-s + (−0.669 − 0.743i)14-s + (0.669 − 0.743i)16-s + (−0.809 + 0.587i)17-s + (0.5 − 0.866i)18-s + (−0.913 − 0.406i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.987 - 0.157i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.987 - 0.157i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08892826562 - 1.121427361i\)
\(L(\frac12)\) \(\approx\) \(0.08892826562 - 1.121427361i\)
\(L(1)\) \(\approx\) \(1.163724778 - 0.3691318611i\)
\(L(1)\) \(\approx\) \(1.163724778 - 0.3691318611i\)

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 \( 1 + (-0.913 + 0.406i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.978 - 0.207i)T \)
13 \( 1 + (-0.978 - 0.207i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (-0.913 - 0.406i)T \)
23 \( 1 + (0.309 - 0.951i)T \)
29 \( 1 + (-0.104 + 0.994i)T \)
31 \( 1 + (-0.913 - 0.406i)T \)
37 \( 1 + (0.669 - 0.743i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + T \)
47 \( 1 + (0.809 + 0.587i)T \)
53 \( 1 + (-0.913 + 0.406i)T \)
59 \( 1 + (0.669 - 0.743i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (-0.913 - 0.406i)T \)
71 \( 1 + (0.104 - 0.994i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (-0.913 - 0.406i)T \)
89 \( 1 + (0.978 - 0.207i)T \)
97 \( 1 + (-0.913 + 0.406i)T \)
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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

−17.73013421174266066075679631613, −17.262161042416576407746870789290, −16.6933247129954374180276786593, −16.00130998959601372769476216438, −15.39062407886892654103604606195, −14.79469315648390742238398701555, −14.05744849027270110522896364165, −13.24060159087126443308020292757, −12.73142750677647642251269834368, −12.127344175196076967975088192585, −11.6532585808015565948767785308, −11.148139719664048024528048876507, −10.19175726789995981160028202112, −9.43923600348309830222065781492, −8.63037397830403887025308270611, −7.57921891426849112879452953173, −6.971284592663684671792372708822, −6.52744677026397339916574031875, −5.70590238714426190059002655892, −5.34828987010263852639876515329, −4.349335996885171981286812222766, −3.95521285817180010001779428988, −2.64354152841192783327659294184, −2.21459318068763947663324412713, −1.31468240715793151743543859878, 0.21929364573445278339789209284, 1.15824526952254498146304836372, 2.119404055622131221539421230670, 3.10020770793029862889416209726, 3.9211983861078568038571759618, 4.40513750053738787252970276292, 4.89752814632914393992596309406, 5.99696530984562789843351220282, 6.36559378355652217751568907527, 6.99095757601968410357975475968, 7.62290373490314494706451698318, 9.017206519092821996310032865920, 9.58497919498951209878850679275, 10.45194202878031298209463689296, 10.94107632071072317699153478944, 11.27799272813573046073819633220, 12.472849907169123900542977920167, 12.57508427824992398608120208421, 13.247626314573602143998821056153, 14.29568662888295571624844037686, 14.69271031432369452850278505260, 15.332749674255100098708844831609, 16.21554594866689910466327285223, 16.625505922638266056210308849693, 17.180423077418858724542776969010

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