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Properties

Label 1-4033-4033.1068-r1-0-0
Degree $1$
Conductor $4033$
Sign $0.578 + 0.815i$
Analytic cond. $433.406$
Root an. cond. $433.406$
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  + (−0.984 − 0.173i)2-s + (0.993 + 0.116i)3-s + (0.939 + 0.342i)4-s + (−0.230 − 0.973i)5-s + (−0.957 − 0.286i)6-s + (−0.286 + 0.957i)7-s + (−0.866 − 0.5i)8-s + (0.973 + 0.230i)9-s + (0.0581 + 0.998i)10-s + (−0.0581 + 0.998i)11-s + (0.893 + 0.448i)12-s + (0.727 − 0.686i)13-s + (0.448 − 0.893i)14-s + (−0.116 − 0.993i)15-s + (0.766 + 0.642i)16-s + (0.984 + 0.173i)17-s + ⋯
L(s)  = 1  + (−0.984 − 0.173i)2-s + (0.993 + 0.116i)3-s + (0.939 + 0.342i)4-s + (−0.230 − 0.973i)5-s + (−0.957 − 0.286i)6-s + (−0.286 + 0.957i)7-s + (−0.866 − 0.5i)8-s + (0.973 + 0.230i)9-s + (0.0581 + 0.998i)10-s + (−0.0581 + 0.998i)11-s + (0.893 + 0.448i)12-s + (0.727 − 0.686i)13-s + (0.448 − 0.893i)14-s + (−0.116 − 0.993i)15-s + (0.766 + 0.642i)16-s + (0.984 + 0.173i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.578 + 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.578 + 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.578 + 0.815i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1068, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ 0.578 + 0.815i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.735827007 + 0.8969041383i\)
\(L(\frac12)\) \(\approx\) \(1.735827007 + 0.8969041383i\)
\(L(1)\) \(\approx\) \(0.9824541236 + 0.02934769300i\)
\(L(1)\) \(\approx\) \(0.9824541236 + 0.02934769300i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.984 - 0.173i)T \)
3 \( 1 + (0.993 + 0.116i)T \)
5 \( 1 + (-0.230 - 0.973i)T \)
7 \( 1 + (-0.286 + 0.957i)T \)
11 \( 1 + (-0.0581 + 0.998i)T \)
13 \( 1 + (0.727 - 0.686i)T \)
17 \( 1 + (0.984 + 0.173i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + (-0.642 + 0.766i)T \)
29 \( 1 + (0.998 - 0.0581i)T \)
31 \( 1 + (-0.802 + 0.597i)T \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 + (-0.642 - 0.766i)T \)
47 \( 1 + (0.686 - 0.727i)T \)
53 \( 1 + (-0.893 - 0.448i)T \)
59 \( 1 + (-0.116 - 0.993i)T \)
61 \( 1 + (0.998 - 0.0581i)T \)
67 \( 1 + (-0.835 - 0.549i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
73 \( 1 + (0.835 + 0.549i)T \)
79 \( 1 + (-0.998 - 0.0581i)T \)
83 \( 1 + (0.973 - 0.230i)T \)
89 \( 1 + (0.802 + 0.597i)T \)
97 \( 1 + (-0.116 + 0.993i)T \)
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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

−18.459449456898166226684417820000, −17.71037193482963705666585746219, −16.62301472237785576995375849816, −16.29004099145812455449133766213, −15.61550518111959745531621804290, −14.67260151099915328250444094141, −14.23194717676002455003765885110, −13.781926378463135602982376974908, −12.74284239277217217332345079721, −11.84591674086072329220551387160, −10.95765555061013048160959438206, −10.489099627931500410586334448819, −9.9293728530056562503974888572, −9.08956701546074748254839159886, −8.35974368532825013106285075950, −7.77118814236577514416656676050, −7.23563880615648768693966796561, −6.34065776740245939723762562869, −6.0719427758993363591100630265, −4.33213663261739189515030321702, −3.59450415834389058430501652253, −3.0251950459431396694079805185, −2.18065222780542873872455248630, −1.26596075787282669431600640757, −0.4221657441287151948974852299, 0.77775862030985794491650463007, 1.69845815375119105678237893464, 2.22510894886652175764672569976, 3.219489636688827910330882433799, 3.81391234311550396190036067393, 4.89841428156553375181828149832, 5.73435985506601279678377384436, 6.685536388975010990878382074139, 7.57686754801180094777407450464, 8.23844105514324174122773571353, 8.59441562259238778899776470772, 9.35292857378880393289781844985, 9.83352001773758985887700173839, 10.52638751806415047037759740956, 11.57303269314872638088247806443, 12.45621283743476152305361708710, 12.61862567488875565468494648033, 13.434116693126600004695695707048, 14.618891598539627980545479460170, 15.21471811619507364674935032667, 15.858677581993743975650107457303, 16.09862747009055800195797148651, 17.17620696521736111266653606145, 17.80812723251482042457927202230, 18.4933967846539993467795028920

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