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Properties

Label 1-6025-6025.729-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.348 + 0.937i$
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  + (−0.809 − 0.587i)2-s + (0.309 − 0.951i)3-s + (0.309 + 0.951i)4-s + (−0.809 + 0.587i)6-s + (0.951 + 0.309i)7-s + (0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + (−0.587 + 0.809i)11-s + 12-s − i·13-s + (−0.587 − 0.809i)14-s + (−0.809 + 0.587i)16-s + (−0.587 + 0.809i)17-s + (0.309 + 0.951i)18-s + (−0.951 − 0.309i)19-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (0.309 − 0.951i)3-s + (0.309 + 0.951i)4-s + (−0.809 + 0.587i)6-s + (0.951 + 0.309i)7-s + (0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + (−0.587 + 0.809i)11-s + 12-s − i·13-s + (−0.587 − 0.809i)14-s + (−0.809 + 0.587i)16-s + (−0.587 + 0.809i)17-s + (0.309 + 0.951i)18-s + (−0.951 − 0.309i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.348 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.348 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.348 + 0.937i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.348 + 0.937i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2150420437 + 0.1494032627i\)
\(L(\frac12)\) \(\approx\) \(0.2150420437 + 0.1494032627i\)
\(L(1)\) \(\approx\) \(0.6218067098 - 0.3126403038i\)
\(L(1)\) \(\approx\) \(0.6218067098 - 0.3126403038i\)

Euler product

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

−17.69779471889155864810992151184, −16.64338828166699319836205268559, −16.32454934008572410907922979945, −15.874055406141744026495015349442, −14.97786690425494265838601378132, −14.457312066001960389298068290704, −13.98241661151342654575541090241, −13.38278214306140515534697054778, −11.95140741411771837055004325673, −11.32847424407079618371799678246, −10.771532316767731449350549748172, −10.300422638480272600984686540513, −9.476096436573107795606225194807, −8.85310435894204371718525617328, −8.213989882226879901137757252936, −7.90064491812186097051065527724, −6.8005715081639707199091799977, −6.184724006770848570220428043415, −5.28202516999118614076057292686, −4.65196691858202786213777501709, −4.17290347857544994498111734353, −2.910929517920199099829021675244, −2.22590295981633779099504382298, −1.34794435983483288361529287337, −0.08712473241408520288895922909, 0.98149870059404843836699536382, 1.87667788246820246120009016912, 2.30239884783931112872722197349, 2.93271704220973277834728984202, 4.06307868572746304105236742265, 4.714715085283775074130772809107, 5.932376226268342244885209853303, 6.43099865913600126232513699115, 7.594577691606355443862440139718, 7.78961275079601054221594275807, 8.3888672838913150244380524427, 9.01250112082866057547238931115, 9.90860524938875566974288527203, 10.62203649657730802712383993168, 11.17224847111123980151312325747, 12.020117190483170966279957640891, 12.42721227114383727748439392902, 13.09075968688120067298585519871, 13.63203749115670241851353910394, 14.62474985407842378029828772514, 15.32309586186304983880757103069, 15.69988229845184164652198899450, 17.088357698954380994296869182976, 17.44144225702024105128967063056, 17.83516152673763809183824680592

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