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Properties

Label 1-6025-6025.169-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.404 + 0.914i$
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  + (−0.406 + 0.913i)2-s + (−0.866 − 0.5i)3-s + (−0.669 − 0.743i)4-s + (0.809 − 0.587i)6-s + (0.777 − 0.629i)7-s + (0.951 − 0.309i)8-s + (0.5 + 0.866i)9-s + (0.358 + 0.933i)11-s + (0.207 + 0.978i)12-s + (−0.998 − 0.0523i)13-s + (0.258 + 0.965i)14-s + (−0.104 + 0.994i)16-s + (0.453 + 0.891i)17-s + (−0.994 + 0.104i)18-s + (−0.998 + 0.0523i)19-s + ⋯
L(s)  = 1  + (−0.406 + 0.913i)2-s + (−0.866 − 0.5i)3-s + (−0.669 − 0.743i)4-s + (0.809 − 0.587i)6-s + (0.777 − 0.629i)7-s + (0.951 − 0.309i)8-s + (0.5 + 0.866i)9-s + (0.358 + 0.933i)11-s + (0.207 + 0.978i)12-s + (−0.998 − 0.0523i)13-s + (0.258 + 0.965i)14-s + (−0.104 + 0.994i)16-s + (0.453 + 0.891i)17-s + (−0.994 + 0.104i)18-s + (−0.998 + 0.0523i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.404 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.404 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.404 + 0.914i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.404 + 0.914i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7896671707 + 0.5143892726i\)
\(L(\frac12)\) \(\approx\) \(0.7896671707 + 0.5143892726i\)
\(L(1)\) \(\approx\) \(0.6521711696 + 0.2005096403i\)
\(L(1)\) \(\approx\) \(0.6521711696 + 0.2005096403i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.406 + 0.913i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (0.777 - 0.629i)T \)
11 \( 1 + (0.358 + 0.933i)T \)
13 \( 1 + (-0.998 - 0.0523i)T \)
17 \( 1 + (0.453 + 0.891i)T \)
19 \( 1 + (-0.998 + 0.0523i)T \)
23 \( 1 + (0.156 + 0.987i)T \)
29 \( 1 + (0.743 + 0.669i)T \)
31 \( 1 + (-0.544 + 0.838i)T \)
37 \( 1 + (-0.358 - 0.933i)T \)
41 \( 1 + (0.951 - 0.309i)T \)
43 \( 1 + (-0.156 + 0.987i)T \)
47 \( 1 + (0.951 - 0.309i)T \)
53 \( 1 + (0.207 - 0.978i)T \)
59 \( 1 + (0.406 - 0.913i)T \)
61 \( 1 + (0.587 - 0.809i)T \)
67 \( 1 + (-0.207 - 0.978i)T \)
71 \( 1 + (-0.0523 - 0.998i)T \)
73 \( 1 + (0.453 - 0.891i)T \)
79 \( 1 + (-0.587 + 0.809i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (0.933 - 0.358i)T \)
97 \( 1 + (-0.978 - 0.207i)T \)
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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

−17.58508087943795955351941203316, −16.9788813666364923128955971476, −16.63252528622637221923996195814, −15.77992091143820756057393087723, −14.91961941358490149286688612456, −14.34641320005029829423680657001, −13.55844028986570676759679868231, −12.57172894709810130751645513393, −12.13428069705107851229467441355, −11.54817936730732406669890212610, −11.10288833707066195462168479857, −10.34135135827462108987068488728, −9.81792703915724814609854668152, −8.94152892593959504650857941487, −8.58067779302985953421122203360, −7.633097144243236864443824809895, −6.84856778017165220176346984766, −5.82516214394639134773854139671, −5.27408007306637966912576107520, −4.39918346699400308331563773941, −4.09914929904999970179164369893, −2.81200496268460618913218951947, −2.42076666657086807584132412435, −1.22727746888522633906799620507, −0.51557107579510490117497163675, 0.66499696079452357247078851542, 1.63574047426392166613792428436, 2.00166083518821920420947801781, 3.73683529216424680602654047587, 4.52146189739359461261010630768, 5.0281128075795146611010683443, 5.65506748510294972439333525648, 6.58655445592778780313719599888, 7.04111902984249222574945839121, 7.65353820556258371648445750137, 8.1517419836411149956330718366, 9.11934908335790536609979911601, 9.92208539496913475872162838346, 10.56200037863071325668630760677, 10.9719784153975343130074781462, 12.01908409166325848874738925753, 12.61545627812640180614830919140, 13.17392354438550105039376206001, 14.260503608773682582925870750201, 14.47563022903557691070461464733, 15.243702327508432908789822693267, 16.02032203883938432154318454352, 16.83440605528973854576824830477, 17.15847705373881188977738722499, 17.79689933654215759651121813852

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