Properties

Degree 1
Conductor 4003
Sign $0.681 + 0.731i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.935 − 0.353i)2-s + (0.837 + 0.546i)3-s + (0.750 − 0.661i)4-s + (−0.968 + 0.250i)5-s + (0.976 + 0.214i)6-s + (−0.619 + 0.785i)7-s + (0.468 − 0.883i)8-s + (0.403 + 0.915i)9-s + (−0.817 + 0.576i)10-s + (0.796 + 0.605i)11-s + (0.989 − 0.143i)12-s + (−0.561 − 0.827i)13-s + (−0.302 + 0.953i)14-s + (−0.947 − 0.319i)15-s + (0.126 − 0.992i)16-s + (0.647 + 0.762i)17-s + ⋯
L(s,χ)  = 1  + (0.935 − 0.353i)2-s + (0.837 + 0.546i)3-s + (0.750 − 0.661i)4-s + (−0.968 + 0.250i)5-s + (0.976 + 0.214i)6-s + (−0.619 + 0.785i)7-s + (0.468 − 0.883i)8-s + (0.403 + 0.915i)9-s + (−0.817 + 0.576i)10-s + (0.796 + 0.605i)11-s + (0.989 − 0.143i)12-s + (−0.561 − 0.827i)13-s + (−0.302 + 0.953i)14-s + (−0.947 − 0.319i)15-s + (0.126 − 0.992i)16-s + (0.647 + 0.762i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.681 + 0.731i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.681 + 0.731i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4003\)
\( \varepsilon \)  =  $0.681 + 0.731i$
motivic weight  =  \(0\)
character  :  $\chi_{4003} (1057, \cdot )$
Sato-Tate  :  $\mu(87)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 4003,\ (0:\ ),\ 0.681 + 0.731i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(3.401272387 + 1.479945865i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(3.401272387 + 1.479945865i\)
\(L(\chi,1)\)  \(\approx\)  \(2.081585190 + 0.2559570584i\)
\(L(1,\chi)\)  \(\approx\)  \(2.081585190 + 0.2559570584i\)

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−18.70059425538278768653480823594, −17.53192293352907221489827169834, −16.63928039719454304124018411922, −16.358743285171264392981442116972, −15.63084109294101660205677123511, −14.62945042305475001235213512603, −14.45224614422764690705778784760, −13.52088449894583908746133270733, −13.23269954721432002777734327102, −12.18561503096327070802021053668, −11.84252217248962169542202354009, −11.248833563091734546317134528272, −9.913216207651413069670885458770, −9.31756605016479488747457342571, −8.33778022790320266181601740435, −7.64597032734879172764356488425, −7.279160302824852641441614813823, −6.5488821084926687750627192257, −5.80121231980172259581710819670, −4.61688790806590778145388413181, −3.90130985195059002348688161906, −3.5397804471111078238011050832, −2.796788761025414701937688012914, −1.696125391985389248643276725908, −0.71839986250002607777784148870, 1.09625523935918020664075347516, 2.25040110524776640923239978888, 3.09728798280895482807707977034, 3.23044017023928206995087421970, 4.31103886426885922407380412511, 4.76879661905140689515425751373, 5.64917304779093497553904930421, 6.710590392711873579563533288085, 7.20056102617244801592705232593, 8.14683233944676524798314072687, 8.89490068491994255499048571624, 9.696234990410001360162698946228, 10.384185412938152775378354676329, 10.95688663994921794514101166394, 11.99023752714494142631063923206, 12.44530073575916714079159247256, 12.91085497063605877625072853652, 14.01393949131291530901435555474, 14.575713251895145897150958475061, 15.18326865610827256937935570303, 15.455374385713952526690918445511, 16.191696574444406663963517370050, 16.89576336095993201669058042019, 18.17011320490900938205752554630, 18.99059599748584809358282266321

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