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Properties

Label 1-6003-6003.1022-r0-0-0
Degree $1$
Conductor $6003$
Sign $-0.635 - 0.772i$
Analytic cond. $27.8778$
Root an. cond. $27.8778$
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  + (−0.976 − 0.215i)2-s + (0.906 + 0.421i)4-s + (−0.115 − 0.993i)5-s + (0.912 + 0.409i)7-s + (−0.794 − 0.607i)8-s + (−0.101 + 0.994i)10-s + (0.155 − 0.987i)11-s + (−0.248 + 0.968i)13-s + (−0.802 − 0.596i)14-s + (0.644 + 0.764i)16-s + (−0.959 + 0.281i)17-s + (0.818 − 0.574i)19-s + (0.314 − 0.949i)20-s + (−0.365 + 0.930i)22-s + (−0.973 + 0.229i)25-s + (0.452 − 0.891i)26-s + ⋯
L(s)  = 1  + (−0.976 − 0.215i)2-s + (0.906 + 0.421i)4-s + (−0.115 − 0.993i)5-s + (0.912 + 0.409i)7-s + (−0.794 − 0.607i)8-s + (−0.101 + 0.994i)10-s + (0.155 − 0.987i)11-s + (−0.248 + 0.968i)13-s + (−0.802 − 0.596i)14-s + (0.644 + 0.764i)16-s + (−0.959 + 0.281i)17-s + (0.818 − 0.574i)19-s + (0.314 − 0.949i)20-s + (−0.365 + 0.930i)22-s + (−0.973 + 0.229i)25-s + (0.452 − 0.891i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.635 - 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.635 - 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $-0.635 - 0.772i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (1022, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ -0.635 - 0.772i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3818557181 - 0.8085394996i\)
\(L(\frac12)\) \(\approx\) \(0.3818557181 - 0.8085394996i\)
\(L(1)\) \(\approx\) \(0.6871704602 - 0.2315437490i\)
\(L(1)\) \(\approx\) \(0.6871704602 - 0.2315437490i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.976 - 0.215i)T \)
5 \( 1 + (-0.115 - 0.993i)T \)
7 \( 1 + (0.912 + 0.409i)T \)
11 \( 1 + (0.155 - 0.987i)T \)
13 \( 1 + (-0.248 + 0.968i)T \)
17 \( 1 + (-0.959 + 0.281i)T \)
19 \( 1 + (0.818 - 0.574i)T \)
31 \( 1 + (0.855 - 0.517i)T \)
37 \( 1 + (0.591 + 0.806i)T \)
41 \( 1 + (0.327 - 0.945i)T \)
43 \( 1 + (-0.855 - 0.517i)T \)
47 \( 1 + (-0.955 - 0.294i)T \)
53 \( 1 + (-0.992 - 0.122i)T \)
59 \( 1 + (0.888 + 0.458i)T \)
61 \( 1 + (0.275 - 0.961i)T \)
67 \( 1 + (-0.155 - 0.987i)T \)
71 \( 1 + (-0.996 + 0.0815i)T \)
73 \( 1 + (0.742 + 0.670i)T \)
79 \( 1 + (0.984 + 0.175i)T \)
83 \( 1 + (-0.942 - 0.333i)T \)
89 \( 1 + (-0.768 - 0.639i)T \)
97 \( 1 + (-0.464 + 0.885i)T \)
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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

−17.89577754399724789826993104703, −17.715451810677130167688335799, −16.7739762136191560823667265193, −15.964657161127992854145184546619, −15.34695098843415774608742852641, −14.73275964092707258490571520941, −14.41045830596345735965159415379, −13.49404528010775337534748262253, −12.51731711521452732977955904318, −11.65458766034046614700081199161, −11.27140275772330391312631631528, −10.55314288345393166043568406928, −9.9746497201706300986902977396, −9.50353627730708610936545835679, −8.371581545469699245906389157017, −7.88820151409705130171769840797, −7.2970596648167533814232845474, −6.77199412448844368312098924810, −5.98250519029807463343252639218, −5.10006507825634207730724309373, −4.352971429926265620781065838616, −3.24806184174605989815539783726, −2.56142941294950247803113044618, −1.80610125571372295770421676122, −0.99225554447717655597872987740, 0.34782028959064610361360538908, 1.26628204151041592680038094827, 1.890669419009377918206559696924, 2.66836296927263071990537131091, 3.68769364086570580408918339259, 4.51439150253916824977174390241, 5.19877404574726182976603753349, 6.09185019240498563189415498521, 6.77892742339732524870425563375, 7.69184444734282668477305307301, 8.34057334268562283032413439615, 8.75091302502832908821636479087, 9.32076016506747354250022694169, 10.0008247764800340519473200427, 11.094638770427142889756144800878, 11.48839358989613792800353367674, 11.86185088751975652107302641420, 12.72447899104853164109571343975, 13.515859560448322798061694532, 14.14093011837201220092540639290, 15.18522994185858005106337684030, 15.65393197968544274174001446710, 16.34331128453318128319924851798, 16.944009164690133079860351247694, 17.41684156978391753010617431978

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