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Properties

Label 1-6025-6025.1602-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.458 - 0.888i$
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  + (0.453 + 0.891i)2-s + (0.453 + 0.891i)3-s + (−0.587 + 0.809i)4-s + (−0.587 + 0.809i)6-s + (0.233 − 0.972i)7-s + (−0.987 − 0.156i)8-s + (−0.587 + 0.809i)9-s + (0.0784 + 0.996i)11-s + (−0.987 − 0.156i)12-s + (0.233 + 0.972i)13-s + (0.972 − 0.233i)14-s + (−0.309 − 0.951i)16-s + (−0.382 + 0.923i)17-s + (−0.987 − 0.156i)18-s + (−0.852 − 0.522i)19-s + ⋯
L(s)  = 1  + (0.453 + 0.891i)2-s + (0.453 + 0.891i)3-s + (−0.587 + 0.809i)4-s + (−0.587 + 0.809i)6-s + (0.233 − 0.972i)7-s + (−0.987 − 0.156i)8-s + (−0.587 + 0.809i)9-s + (0.0784 + 0.996i)11-s + (−0.987 − 0.156i)12-s + (0.233 + 0.972i)13-s + (0.972 − 0.233i)14-s + (−0.309 − 0.951i)16-s + (−0.382 + 0.923i)17-s + (−0.987 − 0.156i)18-s + (−0.852 − 0.522i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.458 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.458 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.458 - 0.888i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (1602, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.458 - 0.888i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.6164855878 + 0.3754695838i\)
\(L(\frac12)\) \(\approx\) \(-0.6164855878 + 0.3754695838i\)
\(L(1)\) \(\approx\) \(0.6794880753 + 0.9299232654i\)
\(L(1)\) \(\approx\) \(0.6794880753 + 0.9299232654i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.453 + 0.891i)T \)
3 \( 1 + (0.453 + 0.891i)T \)
7 \( 1 + (0.233 - 0.972i)T \)
11 \( 1 + (0.0784 + 0.996i)T \)
13 \( 1 + (0.233 + 0.972i)T \)
17 \( 1 + (-0.382 + 0.923i)T \)
19 \( 1 + (-0.852 - 0.522i)T \)
23 \( 1 + (0.382 + 0.923i)T \)
29 \( 1 + (0.707 - 0.707i)T \)
31 \( 1 + (-0.923 - 0.382i)T \)
37 \( 1 + (-0.382 - 0.923i)T \)
41 \( 1 + (-0.891 + 0.453i)T \)
43 \( 1 + (-0.972 + 0.233i)T \)
47 \( 1 + (0.707 + 0.707i)T \)
53 \( 1 + (-0.707 + 0.707i)T \)
59 \( 1 + (0.707 + 0.707i)T \)
61 \( 1 + (-0.707 + 0.707i)T \)
67 \( 1 + (0.707 - 0.707i)T \)
71 \( 1 + (0.923 + 0.382i)T \)
73 \( 1 + (0.233 - 0.972i)T \)
79 \( 1 + (0.987 + 0.156i)T \)
83 \( 1 + (-0.309 - 0.951i)T \)
89 \( 1 + (0.649 - 0.760i)T \)
97 \( 1 - T \)
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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

−17.51630505107937861536687036614, −16.54166578120986830179435203100, −15.53677312153974717075256924955, −15.02308947035127883321444733072, −14.34226228100174107937280916252, −13.79531685380272116376159938121, −13.1457159437786830399825728310, −12.47017826972361546689834174139, −12.14661812197878427170689573259, −11.275916672763249223622514668774, −10.82047948558507149145775345093, −9.90170425431756674608107647798, −9.02772515359750498627588265512, −8.42761707566836223606792192555, −8.2352007982530566715362002704, −6.75471299436050766100347117798, −6.4007255606411449993008286251, −5.3453877897321505177577434656, −5.14600533842828660320908265893, −3.77826541919643686074338625049, −3.170825819405973104627448626262, −2.5904729126572921273588531078, −1.90461947122559898774874692870, −1.064057730891212591717323999073, −0.141694838993192428755005385290, 1.60188018395082133887354128196, 2.440987475371444287808331407736, 3.565666851563208882554719695591, 4.04614561884205119926357817784, 4.54235834647471706299077981482, 5.1255943303939619467102706130, 6.16816804983406872417882147820, 6.81208069768059918076873973694, 7.523318429053350689390212634107, 8.12411068174550829443296846864, 8.952557079352413431287996946542, 9.39800381038031345643327482490, 10.2135830359699376871730251076, 10.930695299767106082209106094552, 11.61203662580721021397797900842, 12.56909566101494568858251030234, 13.330061610729639681941924071960, 13.768627702538968307619475778769, 14.46314916263616016180059395994, 15.07755365465290709489158374071, 15.42005899478843148294746673663, 16.26423861270650830360716290914, 16.890089368084417306830314138503, 17.287915538932551040648039962815, 17.86822947938134463009811915036

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