Properties

Degree 1
Conductor 4003
Sign $0.0948 - 0.995i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.232 − 0.972i)2-s + (−0.968 + 0.250i)3-s + (−0.891 + 0.452i)4-s + (−0.302 + 0.953i)5-s + (0.468 + 0.883i)6-s + (0.197 − 0.980i)7-s + (0.647 + 0.762i)8-s + (0.874 − 0.484i)9-s + (0.997 + 0.0721i)10-s + (−0.994 − 0.108i)11-s + (0.750 − 0.661i)12-s + (−0.161 + 0.986i)13-s + (−0.999 + 0.0361i)14-s + (0.0541 − 0.998i)15-s + (0.590 − 0.806i)16-s + (−0.370 − 0.928i)17-s + ⋯
L(s,χ)  = 1  + (−0.232 − 0.972i)2-s + (−0.968 + 0.250i)3-s + (−0.891 + 0.452i)4-s + (−0.302 + 0.953i)5-s + (0.468 + 0.883i)6-s + (0.197 − 0.980i)7-s + (0.647 + 0.762i)8-s + (0.874 − 0.484i)9-s + (0.997 + 0.0721i)10-s + (−0.994 − 0.108i)11-s + (0.750 − 0.661i)12-s + (−0.161 + 0.986i)13-s + (−0.999 + 0.0361i)14-s + (0.0541 − 0.998i)15-s + (0.590 − 0.806i)16-s + (−0.370 − 0.928i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.0948 - 0.995i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.0948 - 0.995i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4003\)
\( \varepsilon \)  =  $0.0948 - 0.995i$
motivic weight  =  \(0\)
character  :  $\chi_{4003} (945, \cdot )$
Sato-Tate  :  $\mu(87)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 4003,\ (0:\ ),\ 0.0948 - 0.995i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.5459753533 - 0.4964278255i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.5459753533 - 0.4964278255i\)
\(L(\chi,1)\)  \(\approx\)  \(0.5802935372 - 0.2056590550i\)
\(L(1,\chi)\)  \(\approx\)  \(0.5802935372 - 0.2056590550i\)

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

−18.21830601682640215957291128120, −17.9031956381767259251677050086, −17.31837456597746642960695188730, −16.580994869988347894235594819878, −15.93512999733652240642934342489, −15.32377659484724318820118364706, −15.13831557735662883329390065451, −13.64286432733788395014215193763, −13.15162022727472721856368525484, −12.58312846783355190101141196234, −11.92888519400616352808476147978, −11.09456696827511593979434184988, −10.19287614679174983879351331111, −9.61742283997970523427157217124, −8.669801871859741575384019283413, −8.027086681915384959848676515107, −7.62039411832284773143260445286, −6.6074373354946116239819986857, −5.756293947791475289523048419890, −5.306993892216174537158851413111, −4.914313199438528017251945416844, −4.00219502551289800372585941122, −2.68451549576100866854891310435, −1.45406196929848537820281045046, −0.70495422823118944919832265922, 0.44375384385173140305085851518, 1.258607346897596936328045914709, 2.50626150728548066736050143766, 3.08478514400157642184176247331, 4.141558878801909980980330276414, 4.55992118293303952522499575051, 5.31949466341380418860824452432, 6.472444212275936490464503772535, 7.1959919823893045391507891688, 7.6654363299938539730115547771, 8.74924248839712789209430090172, 9.75353856921839693247164796175, 10.232996078056763866381618386784, 10.76926260127101606283478686418, 11.37221010902066118331591659735, 11.840734776342894951258769086191, 12.59787668049302939515302944586, 13.68012582340132820504871926075, 13.86904793106284626926630732709, 14.845001227851310738494214910045, 15.88999662928384535370525250361, 16.328958800612896579512847384761, 17.16803002521799296657990181279, 17.83451519759754223263129853110, 18.31715735501494474551607963428

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