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Properties

Label 1-6145-6145.1007-r0-0-0
Degree 11
Conductor 61456145
Sign 0.563+0.825i0.563 + 0.825i
Analytic cond. 28.537228.5372
Root an. cond. 28.537228.5372
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.675 − 0.737i)2-s + (−0.953 + 0.302i)3-s + (−0.0868 − 0.996i)4-s + (−0.421 + 0.906i)6-s + (0.694 + 0.719i)7-s + (−0.793 − 0.609i)8-s + (0.817 − 0.576i)9-s + (−0.592 + 0.805i)11-s + (0.383 + 0.923i)12-s + (−0.117 + 0.993i)13-s + (0.999 − 0.0255i)14-s + (−0.984 + 0.173i)16-s + (0.994 − 0.102i)17-s + (0.127 − 0.991i)18-s + (0.369 − 0.929i)19-s + ⋯
L(s)  = 1  + (0.675 − 0.737i)2-s + (−0.953 + 0.302i)3-s + (−0.0868 − 0.996i)4-s + (−0.421 + 0.906i)6-s + (0.694 + 0.719i)7-s + (−0.793 − 0.609i)8-s + (0.817 − 0.576i)9-s + (−0.592 + 0.805i)11-s + (0.383 + 0.923i)12-s + (−0.117 + 0.993i)13-s + (0.999 − 0.0255i)14-s + (−0.984 + 0.173i)16-s + (0.994 − 0.102i)17-s + (0.127 − 0.991i)18-s + (0.369 − 0.929i)19-s + ⋯

Functional equation

Λ(s)=(6145s/2 ΓR(s) L(s)=((0.563+0.825i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.563 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(6145s/2 ΓR(s) L(s)=((0.563+0.825i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.563 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 61456145    =    5⋅12295 \cdot 1229
Sign: 0.563+0.825i0.563 + 0.825i
Analytic conductor: 28.537228.5372
Root analytic conductor: 28.537228.5372
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ6145(1007,⋅)\chi_{6145} (1007, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 6145, (0: ), 0.563+0.825i)(1,\ 6145,\ (0:\ ),\ 0.563 + 0.825i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.257823612+0.6643465070i1.257823612 + 0.6643465070i
L(12)L(\frac12) ≈\approx 1.257823612+0.6643465070i1.257823612 + 0.6643465070i
L(1)L(1) ≈\approx 1.084070606−0.1626697351i1.084070606 - 0.1626697351i
L(1)L(1) ≈\approx 1.084070606−0.1626697351i1.084070606 - 0.1626697351i

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1 1
1229 1 1
good2 1+(0.675−0.737i)T 1 + (0.675 - 0.737i)T
3 1+(−0.953+0.302i)T 1 + (-0.953 + 0.302i)T
7 1+(0.694+0.719i)T 1 + (0.694 + 0.719i)T
11 1+(−0.592+0.805i)T 1 + (-0.592 + 0.805i)T
13 1+(−0.117+0.993i)T 1 + (-0.117 + 0.993i)T
17 1+(0.994−0.102i)T 1 + (0.994 - 0.102i)T
19 1+(0.369−0.929i)T 1 + (0.369 - 0.929i)T
23 1+(−0.956−0.292i)T 1 + (-0.956 - 0.292i)T
29 1+(0.567+0.823i)T 1 + (0.567 + 0.823i)T
31 1+(0.999+0.0409i)T 1 + (0.999 + 0.0409i)T
37 1+(−0.946−0.321i)T 1 + (-0.946 - 0.321i)T
41 1+(0.364+0.931i)T 1 + (0.364 + 0.931i)T
43 1+(0.744+0.668i)T 1 + (0.744 + 0.668i)T
47 1+(−0.998+0.0562i)T 1 + (-0.998 + 0.0562i)T
53 1+(−0.927−0.374i)T 1 + (-0.927 - 0.374i)T
59 1+(0.983+0.183i)T 1 + (0.983 + 0.183i)T
61 1+(0.448+0.893i)T 1 + (0.448 + 0.893i)T
67 1+(−0.985−0.168i)T 1 + (-0.985 - 0.168i)T
71 1+(−0.0919−0.995i)T 1 + (-0.0919 - 0.995i)T
73 1+(0.957−0.287i)T 1 + (0.957 - 0.287i)T
79 1+(−0.793+0.609i)T 1 + (-0.793 + 0.609i)T
83 1+(0.981+0.193i)T 1 + (0.981 + 0.193i)T
89 1+(−0.0766−0.997i)T 1 + (-0.0766 - 0.997i)T
97 1+(0.435−0.900i)T 1 + (0.435 - 0.900i)T
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   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

−17.43396523085177131279777555757, −16.99880783890510105128233950294, −16.01707817129238113219078651955, −15.9663910537819438981202134079, −14.99219717569015940722306539898, −14.05835538502103457154010523998, −13.817689066875419438706860596078, −13.04780399996632585326489014006, −12.290220663796544477998051679221, −11.87799890266031408987010009066, −11.115152194506670028358428677040, −10.35191682160008202888248856193, −9.907386800545748363479245473, −8.40881317434702711520400657648, −7.88933816230474577254443897895, −7.63104531681381486549009741384, −6.67333089409524473459762045971, −5.91328398269242628799285499137, −5.46243841259996301720329874009, −4.9195331052434017541044958160, −4.00377427596107968202207715943, −3.41221022496034867818085243707, −2.37623514719712399811761282851, −1.27402169158581324222460512670, −0.35934546317284591974314875020, 1.02323730364113882259367644448, 1.748918711232539959487903865039, 2.51956354224896341884169074917, 3.37024612279674533626046301569, 4.48915145396029118513817623647, 4.71795586120714863934764165808, 5.34716504581673830133871271728, 6.09240645639103614018892896904, 6.75892204461836178583127976138, 7.58960444395279222195762890801, 8.63577067326540441722755280260, 9.52349525461983492302957783322, 9.92452968112043653706841862657, 10.68561410400574837972239047810, 11.33601301641933544407813227557, 11.93522321536349502129551423567, 12.26215583329767049683084866600, 12.94085917504255484349064207962, 13.85538006216048229436076909976, 14.54355693164025516940010414955, 15.02166204594235580919591203551, 15.9729639185794486755054390050, 16.08910210778280027548843144217, 17.305424325432469632027848273857, 18.08049012151832151859892227751

Graph of the ZZ-function along the critical line