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Properties

Label 1-6041-6041.104-r0-0-0
Degree $1$
Conductor $6041$
Sign $-0.998 - 0.0522i$
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  + (−0.290 − 0.956i)2-s + (−0.472 + 0.881i)3-s + (−0.830 + 0.556i)4-s + (−0.453 − 0.891i)5-s + (0.980 + 0.195i)6-s + (0.774 + 0.632i)8-s + (−0.553 − 0.832i)9-s + (−0.721 + 0.692i)10-s + (0.318 − 0.947i)11-s + (−0.0982 − 0.995i)12-s + (0.547 − 0.836i)13-s + (0.999 + 0.0218i)15-s + (0.380 − 0.924i)16-s + (−0.241 + 0.970i)17-s + (−0.635 + 0.771i)18-s + (−0.170 + 0.985i)19-s + ⋯
L(s)  = 1  + (−0.290 − 0.956i)2-s + (−0.472 + 0.881i)3-s + (−0.830 + 0.556i)4-s + (−0.453 − 0.891i)5-s + (0.980 + 0.195i)6-s + (0.774 + 0.632i)8-s + (−0.553 − 0.832i)9-s + (−0.721 + 0.692i)10-s + (0.318 − 0.947i)11-s + (−0.0982 − 0.995i)12-s + (0.547 − 0.836i)13-s + (0.999 + 0.0218i)15-s + (0.380 − 0.924i)16-s + (−0.241 + 0.970i)17-s + (−0.635 + 0.771i)18-s + (−0.170 + 0.985i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0522i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6041\)    =    \(7 \cdot 863\)
Sign: $-0.998 - 0.0522i$
Analytic conductor: \(28.0542\)
Root analytic conductor: \(28.0542\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6041} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6041,\ (0:\ ),\ -0.998 - 0.0522i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01048735526 - 0.4009548148i\)
\(L(\frac12)\) \(\approx\) \(0.01048735526 - 0.4009548148i\)
\(L(1)\) \(\approx\) \(0.5653411814 - 0.2255027512i\)
\(L(1)\) \(\approx\) \(0.5653411814 - 0.2255027512i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
863 \( 1 \)
good2 \( 1 + (-0.290 - 0.956i)T \)
3 \( 1 + (-0.472 + 0.881i)T \)
5 \( 1 + (-0.453 - 0.891i)T \)
11 \( 1 + (0.318 - 0.947i)T \)
13 \( 1 + (0.547 - 0.836i)T \)
17 \( 1 + (-0.241 + 0.970i)T \)
19 \( 1 + (-0.170 + 0.985i)T \)
23 \( 1 + (-0.433 + 0.901i)T \)
29 \( 1 + (0.999 + 0.0145i)T \)
31 \( 1 + (-0.929 - 0.370i)T \)
37 \( 1 + (-0.854 + 0.519i)T \)
41 \( 1 + (0.504 + 0.863i)T \)
43 \( 1 + (-0.0182 - 0.999i)T \)
47 \( 1 + (-0.902 + 0.430i)T \)
53 \( 1 + (-0.359 - 0.933i)T \)
59 \( 1 + (0.999 + 0.00728i)T \)
61 \( 1 + (0.0327 + 0.999i)T \)
67 \( 1 + (0.105 - 0.994i)T \)
71 \( 1 + (0.911 + 0.410i)T \)
73 \( 1 + (-0.778 - 0.627i)T \)
79 \( 1 + (-0.127 - 0.991i)T \)
83 \( 1 + (0.0109 - 0.999i)T \)
89 \( 1 + (0.783 + 0.621i)T \)
97 \( 1 + (-0.177 - 0.984i)T \)
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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

−17.953659193289111409420798149016, −17.58270072320690323295939155590, −16.75542788794581873857185666451, −15.934898439621694073104443081302, −15.70416257271898337320343562884, −14.53776769219238307591615468445, −14.304323643776799616232033527164, −13.643814285515049008169453752573, −12.82740973790242841518646356904, −12.14224144563267149708737712497, −11.37304860601424780724347766950, −10.820881602446089505900563686968, −10.05587598073351221783168013306, −9.17405137912066871160322422286, −8.54052282952007769679390289208, −7.7330146321783622876117922423, −7.04350592999673165246132815864, −6.75980596323560296966093345413, −6.27399628118475406642916974407, −5.22676802858499480168502095243, −4.58852573844327925413239614093, −3.84008332888402381321010731272, −2.61662192919631185859882059067, −1.91494315410828928036127482234, −0.86619090263516102137889308676, 0.16881587359525219162652653736, 1.07256182285462103898781679227, 1.79436546222488788288043074210, 3.22076016206148809430934369569, 3.60007034468907226891290825905, 4.14299249204192392427991150537, 5.00148731416883775087780398343, 5.63912877587359665934702518776, 6.269648401540143636114407941120, 7.71425226371563285495255735247, 8.38770933191079836266270495544, 8.70581567377090800214035097256, 9.4983980107426631444432844139, 10.243042609161117662315647315970, 10.70204341675524860197865167162, 11.51937193696090703527232870567, 11.83275695655264097937567872100, 12.682537709594333918406415785711, 13.15875054568794975561683339856, 14.01673641914014174344432724879, 14.80041571149533751114601713807, 15.62580899846740429538420005046, 16.2783996849548430892280152786, 16.6926742999554058554810707629, 17.46114938221305559716310945517

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