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Properties

Label 1-6025-6025.1296-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.170 - 0.985i$
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 + (−0.809 − 0.587i)3-s + (0.309 + 0.951i)4-s + (0.309 + 0.951i)6-s + (0.309 + 0.951i)7-s + (0.309 − 0.951i)8-s + (0.309 + 0.951i)9-s + (−0.809 − 0.587i)11-s + (0.309 − 0.951i)12-s + (0.309 − 0.951i)13-s + (0.309 − 0.951i)14-s + (−0.809 + 0.587i)16-s + 17-s + (0.309 − 0.951i)18-s + (0.309 − 0.951i)19-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (−0.809 − 0.587i)3-s + (0.309 + 0.951i)4-s + (0.309 + 0.951i)6-s + (0.309 + 0.951i)7-s + (0.309 − 0.951i)8-s + (0.309 + 0.951i)9-s + (−0.809 − 0.587i)11-s + (0.309 − 0.951i)12-s + (0.309 − 0.951i)13-s + (0.309 − 0.951i)14-s + (−0.809 + 0.587i)16-s + 17-s + (0.309 − 0.951i)18-s + (0.309 − 0.951i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.170 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.170 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.170 - 0.985i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (1296, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.170 - 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8655683181 - 0.7284069772i\)
\(L(\frac12)\) \(\approx\) \(0.8655683181 - 0.7284069772i\)
\(L(1)\) \(\approx\) \(0.6411835214 - 0.2786816204i\)
\(L(1)\) \(\approx\) \(0.6411835214 - 0.2786816204i\)

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 \( 1 + (-0.809 - 0.587i)T \)
7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (-0.809 - 0.587i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
17 \( 1 + T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + T \)
29 \( 1 + T \)
31 \( 1 + T \)
37 \( 1 + T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (0.309 - 0.951i)T \)
47 \( 1 + T \)
53 \( 1 + T \)
59 \( 1 + T \)
61 \( 1 + T \)
67 \( 1 + T \)
71 \( 1 + T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (0.309 - 0.951i)T \)
83 \( 1 + (-0.809 + 0.587i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
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

−17.7296847838219162130210208534, −17.0339465125603776242306729237, −16.65847468270498802567033509600, −16.10796721136833117335611071414, −15.46888449744165264126031793776, −14.70010628776725630412695687080, −14.21530744778748270308047202971, −13.39188558232701965861055149888, −12.428565914538616253068226686742, −11.61451008634750057889094507810, −11.135024118218268516404482083485, −10.250052528479354696525200609551, −10.072291096218229821752881301965, −9.40291003851414649353053546153, −8.39767401258162452920154699845, −7.80325604315113975981744244338, −7.030483039530016240809241884924, −6.53289700385073273795458532961, −5.70179933919965926922535919095, −4.98547823746939270824300199774, −4.47274209813218853625429342010, −3.595169494255059549251900799287, −2.44950608125844552607357988522, −1.24550380770040203170105688741, −0.86287645711828909220282459031, 0.73854652422298453183723902332, 0.98151762336325846184832490038, 2.347870894158984452803350409018, 2.65113634084578754648101649337, 3.53038740792895827239310634709, 4.8218009951303339722306666127, 5.38708399782986022560320039391, 6.034071335944158884058322063326, 6.96828329054371315403155061600, 7.5713309146804291196122138257, 8.43694917201302109877493485263, 8.57242572817745813245644622355, 9.80401588925146144457177586350, 10.347841227194096691175211853236, 11.03657441687273449410585035060, 11.54284925830155548781528064244, 12.16652830097495949262145696259, 12.78090031930893932808139818368, 13.2892487965951172706863267620, 14.06638299466685602879641503332, 15.435026664990387540696549528044, 15.619560421449821972809329938125, 16.4216419667960394443730097469, 17.23696665363200144889411297388, 17.58632295235823432056115003316

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