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Properties

Label 1-6041-6041.125-r0-0-0
Degree $1$
Conductor $6041$
Sign $-0.116 + 0.993i$
Analytic cond. $28.0542$
Root an. cond. $28.0542$
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  + (0.929 + 0.370i)2-s + (0.914 + 0.403i)3-s + (0.726 + 0.687i)4-s + (−0.999 − 0.0218i)5-s + (0.700 + 0.713i)6-s + (0.420 + 0.907i)8-s + (0.674 + 0.738i)9-s + (−0.920 − 0.390i)10-s + (−0.979 − 0.202i)11-s + (0.386 + 0.922i)12-s + (0.559 − 0.828i)13-s + (−0.905 − 0.423i)15-s + (0.0546 + 0.998i)16-s + (−0.170 + 0.985i)17-s + (0.353 + 0.935i)18-s + (0.959 + 0.280i)19-s + ⋯
L(s)  = 1  + (0.929 + 0.370i)2-s + (0.914 + 0.403i)3-s + (0.726 + 0.687i)4-s + (−0.999 − 0.0218i)5-s + (0.700 + 0.713i)6-s + (0.420 + 0.907i)8-s + (0.674 + 0.738i)9-s + (−0.920 − 0.390i)10-s + (−0.979 − 0.202i)11-s + (0.386 + 0.922i)12-s + (0.559 − 0.828i)13-s + (−0.905 − 0.423i)15-s + (0.0546 + 0.998i)16-s + (−0.170 + 0.985i)17-s + (0.353 + 0.935i)18-s + (0.959 + 0.280i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6041\)    =    \(7 \cdot 863\)
Sign: $-0.116 + 0.993i$
Analytic conductor: \(28.0542\)
Root analytic conductor: \(28.0542\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6041} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6041,\ (0:\ ),\ -0.116 + 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.850946841 + 3.205026395i\)
\(L(\frac12)\) \(\approx\) \(2.850946841 + 3.205026395i\)
\(L(1)\) \(\approx\) \(1.988047730 + 1.022228135i\)
\(L(1)\) \(\approx\) \(1.988047730 + 1.022228135i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
863 \( 1 \)
good2 \( 1 + (0.929 + 0.370i)T \)
3 \( 1 + (0.914 + 0.403i)T \)
5 \( 1 + (-0.999 - 0.0218i)T \)
11 \( 1 + (-0.979 - 0.202i)T \)
13 \( 1 + (0.559 - 0.828i)T \)
17 \( 1 + (-0.170 + 0.985i)T \)
19 \( 1 + (0.959 + 0.280i)T \)
23 \( 1 + (0.896 - 0.443i)T \)
29 \( 1 + (0.957 + 0.287i)T \)
31 \( 1 + (-0.269 - 0.962i)T \)
37 \( 1 + (-0.0619 + 0.998i)T \)
41 \( 1 + (0.413 - 0.910i)T \)
43 \( 1 + (0.934 - 0.356i)T \)
47 \( 1 + (-0.861 - 0.507i)T \)
53 \( 1 + (0.472 - 0.881i)T \)
59 \( 1 + (-0.989 - 0.145i)T \)
61 \( 1 + (-0.792 + 0.609i)T \)
67 \( 1 + (0.516 - 0.856i)T \)
71 \( 1 + (-0.565 + 0.824i)T \)
73 \( 1 + (0.547 + 0.836i)T \)
79 \( 1 + (0.830 + 0.556i)T \)
83 \( 1 + (0.976 + 0.216i)T \)
89 \( 1 + (0.663 + 0.748i)T \)
97 \( 1 + (-0.908 + 0.416i)T \)
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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

−17.952526455303372992147296087797, −16.35812321507048121172561299105, −15.98054383234903237453256083715, −15.47882431299299448941340235909, −14.85920575354950024942832216800, −14.0049883389598600999149822451, −13.76173374137434772748894731520, −12.915572627858560939806046508142, −12.407297084958743144495362420215, −11.709424108533076452577152425805, −11.12831524154556160635414813682, −10.44004078488223095559169960999, −9.39690089726521029973427802757, −9.00241633725791501896488565701, −7.867747999853162689507048827761, −7.43569465021096514576405692638, −6.87765173372934819853589864140, −6.06044716176374243787684505750, −4.85171803161763779116943038918, −4.61852782699981403437341818767, −3.60018564965937740335603213064, −3.045685914371923389266386239409, −2.55969963777745912392948171814, −1.50613228672325562802512142118, −0.74901137306272976118419969168, 1.04086833162276089198260729772, 2.26011561022333119990002737774, 2.99248463172413171117084803895, 3.480047506936985043486395291362, 4.0848439506933131727259652230, 4.90934299857288697466205600693, 5.418229320409035201748363129040, 6.41861271181678903893645606213, 7.28317811360535941551263524048, 7.893300866991397565435051742787, 8.26776258826001000676290736232, 8.89154802221961115186035701158, 10.104096821276538183028449245225, 10.76266668208676572886558706977, 11.18603484948708000596833796377, 12.26221676819788987085164507458, 12.746557392157065788277833531803, 13.382290064260424020513274738230, 13.93933379004114920423106242390, 14.82125651693698218220490577871, 15.251204400900681459098913853774, 15.663948340209325351546733211159, 16.249444105681101335205242134836, 16.84751877837444164380600040444, 17.90679471036867033801258744117

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