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Properties

Label 1-6043-6043.74-r1-0-0
Degree $1$
Conductor $6043$
Sign $0.975 - 0.221i$
Analytic cond. $649.410$
Root an. cond. $649.410$
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  + (−0.412 − 0.911i)2-s + (0.474 + 0.880i)3-s + (−0.660 + 0.751i)4-s + (0.173 − 0.984i)5-s + (0.606 − 0.795i)6-s + (0.597 − 0.802i)7-s + (0.956 + 0.292i)8-s + (−0.549 + 0.835i)9-s + (−0.968 + 0.248i)10-s + (−0.350 + 0.936i)11-s + (−0.974 − 0.224i)12-s + (−0.445 + 0.895i)13-s + (−0.976 − 0.213i)14-s + (0.949 − 0.314i)15-s + (−0.128 − 0.991i)16-s + (0.998 − 0.0540i)17-s + ⋯
L(s)  = 1  + (−0.412 − 0.911i)2-s + (0.474 + 0.880i)3-s + (−0.660 + 0.751i)4-s + (0.173 − 0.984i)5-s + (0.606 − 0.795i)6-s + (0.597 − 0.802i)7-s + (0.956 + 0.292i)8-s + (−0.549 + 0.835i)9-s + (−0.968 + 0.248i)10-s + (−0.350 + 0.936i)11-s + (−0.974 − 0.224i)12-s + (−0.445 + 0.895i)13-s + (−0.976 − 0.213i)14-s + (0.949 − 0.314i)15-s + (−0.128 − 0.991i)16-s + (0.998 − 0.0540i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6043 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.975 - 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6043 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.975 - 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6043\)
Sign: $0.975 - 0.221i$
Analytic conductor: \(649.410\)
Root analytic conductor: \(649.410\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6043} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6043,\ (1:\ ),\ 0.975 - 0.221i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.176306654 - 0.2443518177i\)
\(L(\frac12)\) \(\approx\) \(2.176306654 - 0.2443518177i\)
\(L(1)\) \(\approx\) \(1.021447312 - 0.2061270700i\)
\(L(1)\) \(\approx\) \(1.021447312 - 0.2061270700i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad6043 \( 1 \)
good2 \( 1 + (-0.412 - 0.911i)T \)
3 \( 1 + (0.474 + 0.880i)T \)
5 \( 1 + (0.173 - 0.984i)T \)
7 \( 1 + (0.597 - 0.802i)T \)
11 \( 1 + (-0.350 + 0.936i)T \)
13 \( 1 + (-0.445 + 0.895i)T \)
17 \( 1 + (0.998 - 0.0540i)T \)
19 \( 1 + (0.974 - 0.223i)T \)
23 \( 1 + (0.135 + 0.990i)T \)
29 \( 1 + (0.0244 - 0.999i)T \)
31 \( 1 + (-0.843 + 0.536i)T \)
37 \( 1 + (0.515 - 0.857i)T \)
41 \( 1 + (-0.610 + 0.792i)T \)
43 \( 1 + (-0.100 - 0.994i)T \)
47 \( 1 + (0.0805 + 0.996i)T \)
53 \( 1 + (-0.413 - 0.910i)T \)
59 \( 1 + (0.246 + 0.969i)T \)
61 \( 1 + (0.217 + 0.976i)T \)
67 \( 1 + (0.991 + 0.130i)T \)
71 \( 1 + (0.175 - 0.984i)T \)
73 \( 1 + (-0.814 - 0.580i)T \)
79 \( 1 + (-0.503 - 0.863i)T \)
83 \( 1 + (0.679 + 0.733i)T \)
89 \( 1 + (0.996 - 0.0851i)T \)
97 \( 1 + (0.526 + 0.850i)T \)
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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

−17.69558179656204704692721079437, −17.08161992958871043949371103997, −16.22921876119996008718000154567, −15.44445336913762753131381933623, −14.824914145634167647792081567374, −14.38360926606826375109281411518, −13.98208874809682259587968887070, −13.10777717614511244271395231979, −12.47738273673202986041133522835, −11.56738868449117687517859858382, −10.90885710471698271987321160890, −10.09472933835009025668930371606, −9.45329985042578209051545411286, −8.56814581962448742218861348498, −8.07535901541536262703806083130, −7.59329922854437232447394489819, −6.90757322996628226664856478587, −6.12223516991796207409843969839, −5.583384509306142216629935364875, −5.072361076617427285970656622335, −3.56536418556012395609513573677, −3.004398273109924875637657721758, −2.210941402065053959224545875504, −1.29719218598802137521851557776, −0.49324567382620592168297284346, 0.57149731150590160862698551399, 1.54615721718322642432259262566, 2.00342212838133097499897518910, 3.024149412813008427015688841852, 3.87660586168628126203968362581, 4.36106040607659596973293562781, 5.02615981195241844389940434078, 5.4740002633909048270447326795, 7.21594767440838977895356147656, 7.69686887435624076425360985382, 8.24360994220367787808857076592, 9.32785216635360049539819477899, 9.412463703986096030547420151905, 10.1104090881123045020080214355, 10.728363060704297355771900239876, 11.73608382999924613460075069125, 11.883849438413391606397977505769, 13.00416862744758828304692016212, 13.50082437236485285403678280534, 14.16213451057626723353779672274, 14.71683892027207619496612892427, 15.74507389693409247249397292945, 16.459826375807839095367985451183, 16.81417931248391519388358461263, 17.56541096981003191708901496929

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