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Properties

Label 1-6025-6025.4522-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.666 + 0.745i$
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  + (0.156 + 0.987i)2-s + (−0.891 + 0.453i)3-s + (−0.951 + 0.309i)4-s + (−0.587 − 0.809i)6-s + (0.649 − 0.760i)7-s + (−0.453 − 0.891i)8-s + (0.587 − 0.809i)9-s + (0.852 − 0.522i)11-s + (0.707 − 0.707i)12-s + (0.923 − 0.382i)13-s + (0.852 + 0.522i)14-s + (0.809 − 0.587i)16-s + (0.972 + 0.233i)17-s + (0.891 + 0.453i)18-s + (−0.649 + 0.760i)19-s + ⋯
L(s)  = 1  + (0.156 + 0.987i)2-s + (−0.891 + 0.453i)3-s + (−0.951 + 0.309i)4-s + (−0.587 − 0.809i)6-s + (0.649 − 0.760i)7-s + (−0.453 − 0.891i)8-s + (0.587 − 0.809i)9-s + (0.852 − 0.522i)11-s + (0.707 − 0.707i)12-s + (0.923 − 0.382i)13-s + (0.852 + 0.522i)14-s + (0.809 − 0.587i)16-s + (0.972 + 0.233i)17-s + (0.891 + 0.453i)18-s + (−0.649 + 0.760i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.666 + 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.666 + 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.666 + 0.745i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (4522, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.666 + 0.745i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6394198915 + 1.429807543i\)
\(L(\frac12)\) \(\approx\) \(0.6394198915 + 1.429807543i\)
\(L(1)\) \(\approx\) \(0.8001671102 + 0.5739286182i\)
\(L(1)\) \(\approx\) \(0.8001671102 + 0.5739286182i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.156 + 0.987i)T \)
3 \( 1 + (-0.891 + 0.453i)T \)
7 \( 1 + (0.649 - 0.760i)T \)
11 \( 1 + (0.852 - 0.522i)T \)
13 \( 1 + (0.923 - 0.382i)T \)
17 \( 1 + (0.972 + 0.233i)T \)
19 \( 1 + (-0.649 + 0.760i)T \)
23 \( 1 + (-0.0784 + 0.996i)T \)
29 \( 1 + (0.987 + 0.156i)T \)
31 \( 1 + (-0.233 + 0.972i)T \)
37 \( 1 + (-0.649 + 0.760i)T \)
41 \( 1 + (0.891 + 0.453i)T \)
43 \( 1 + (0.760 - 0.649i)T \)
47 \( 1 + (-0.156 + 0.987i)T \)
53 \( 1 + (0.987 + 0.156i)T \)
59 \( 1 + (0.453 + 0.891i)T \)
61 \( 1 + (-0.453 + 0.891i)T \)
67 \( 1 + (-0.156 - 0.987i)T \)
71 \( 1 + (-0.852 - 0.522i)T \)
73 \( 1 + (0.923 + 0.382i)T \)
79 \( 1 + (-0.987 + 0.156i)T \)
83 \( 1 + (0.309 + 0.951i)T \)
89 \( 1 + (0.233 - 0.972i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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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

−17.7324741369635644079052292924, −17.06405555000796189623304847226, −16.307076019823533035975489575727, −15.46618447662248362712632479895, −14.59467207984775356396685327267, −14.14894342655013608107046303107, −13.32293929473969234749053948221, −12.555868727963188064211811072691, −12.19501910033874692557680529391, −11.51960068006919205212432080698, −11.10643245492385598281152510662, −10.40312727690847824176285302988, −9.62387846931369897220645091631, −8.83603399571814434083310764690, −8.317688016207570364163842641918, −7.35943132995912061863640264603, −6.43392511610043732505768567676, −5.86490032191327256383356479426, −5.145461828263942719001631300004, −4.4211389572052377299352915310, −3.910613497805014229111290585491, −2.64821148651330246103809670860, −2.04033524608404829665158080905, −1.33543815705097856956431543523, −0.5448225553078735743074924422, 1.09300911659611522842914015070, 1.22334014620385559148934359393, 3.27190066167371620532172153907, 3.8091049670202373468319093634, 4.34688564005031492958858317132, 5.18781901983462910224842390853, 5.836867237195402324760711152750, 6.30623073952288249168723349380, 7.09097653182886729493229879625, 7.79918531009126309822365837898, 8.52928070365122578679753846899, 9.15099225137274079114469153425, 10.1545944489800248873720109577, 10.51027599956674187172687249077, 11.353418095805818324825438587575, 12.08961342120026872359578932984, 12.65287717399203779681688484646, 13.63619909172330051490344713301, 14.08452444137199032426588380143, 14.767738140577892394854621723276, 15.39501764252959458956143217964, 16.23618319770438827127758621955, 16.516857437631222524516173402395, 17.230309049908656384249198635372, 17.66222371560823380068018690698

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