↑

Properties

Label 1-703-703.322-r1-0-0
Degree 11
Conductor 703703
Sign 0.989−0.146i0.989 - 0.146i
Analytic cond. 75.547875.5478
Root an. cond. 75.547875.5478
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.5 − 0.866i)5-s + 6-s + (−0.5 − 0.866i)7-s − 8-s + (−0.5 + 0.866i)9-s − 10-s + 11-s + (0.5 − 0.866i)12-s + (0.5 + 0.866i)13-s − 14-s + (0.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.5 − 0.866i)5-s + 6-s + (−0.5 − 0.866i)7-s − 8-s + (−0.5 + 0.866i)9-s − 10-s + 11-s + (0.5 − 0.866i)12-s + (0.5 + 0.866i)13-s − 14-s + (0.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯

Functional equation

Λ(s)=(703s/2 ΓR(s+1) L(s)=((0.989−0.146i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 703 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.989 - 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(703s/2 ΓR(s+1) L(s)=((0.989−0.146i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 703 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.989 - 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 703703    =    19⋅3719 \cdot 37
Sign: 0.989−0.146i0.989 - 0.146i
Analytic conductor: 75.547875.5478
Root analytic conductor: 75.547875.5478
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ703(322,⋅)\chi_{703} (322, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 703, (1: ), 0.989−0.146i)(1,\ 703,\ (1:\ ),\ 0.989 - 0.146i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.214886383−0.1632016681i2.214886383 - 0.1632016681i
L(12)L(\frac12) ≈\approx 2.214886383−0.1632016681i2.214886383 - 0.1632016681i
L(1)L(1) ≈\approx 1.259204728−0.3940463242i1.259204728 - 0.3940463242i
L(1)L(1) ≈\approx 1.259204728−0.3940463242i1.259204728 - 0.3940463242i

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad19 1 1
37 1 1
good2 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
3 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
5 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
7 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
11 1+T 1 + T
13 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
17 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
23 1+T 1 + T
29 1−T 1 - T
31 1−T 1 - T
41 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
43 1+T 1 + T
47 1+T 1 + T
53 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
59 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
61 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
67 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
71 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
73 1+T 1 + T
79 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
83 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
89 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
97 1−T 1 - T
show more
show less
   L(s)=∏p (1−αp p−s)−1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−22.61049227841956367820708203093, −22.10874869746563130087303465812, −20.860996556928325740139598539979, −19.886622786815281045619134554238, −18.98672458070246060816261835542, −18.34193158655769974813297647461, −17.72514354889784778725501657846, −16.61730173146363532798549123137, −15.48089753368192091367867968675, −15.09951565992742425291175371615, −14.27595093089527776679044865011, −13.47776715005363374570717701767, −12.610659658001657245116704904539, −11.94071132845257801324873851566, −11.02390967598203215481627309868, −9.219460998242197856351722461296, −8.87791673070178492339038002074, −7.64557222229898321439398678617, −7.09730778703760841653938830852, −6.26996798101649429903055345212, −5.546059699636624839135843341143, −3.92921607927366257554929795183, −3.1879410952263701131096831092, −2.38295305887428668072140634465, −0.50418281436276016260235677445, 0.90566247410159515378412287751, 1.98257334613412273442782024232, 3.54386114478448621132613669084, 3.93317647871461973967655656542, 4.587231001543304192843002877990, 5.730169301424559512842994494126, 6.95650565150755365860194411645, 8.401082546505326943239213136110, 9.2045190274157466961860281538, 9.64526612920535223488142008854, 10.99884815577992319106323148854, 11.23909172416687176840950211083, 12.58043103680751352320892645497, 13.22658447068052320108026061613, 14.091887490664949338259732275651, 14.83595532908426994018301282205, 15.75759519568350383096456625519, 16.64779795215088020764353064102, 17.2310792150063373671219116651, 18.89730400361114864458022748153, 19.53201725337332961534988402214, 20.06840170083916296862869961721, 20.71478213494739708449041452423, 21.46367386155523473085096530480, 22.277037535822263267067001302614

Graph of the ZZ-function along the critical line