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Properties

Label 2-2793-2793.584-c0-0-0
Degree 22
Conductor 27932793
Sign 0.252+0.967i0.252 + 0.967i
Analytic cond. 1.393881.39388
Root an. cond. 1.180631.18063
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.797 − 0.603i)3-s + (−0.878 − 0.478i)4-s + (0.995 − 0.0995i)7-s + (0.270 − 0.962i)9-s + (−0.988 + 0.149i)12-s + (0.145 + 0.201i)13-s + (0.542 + 0.840i)16-s + (0.900 − 0.433i)19-s + (0.733 − 0.680i)21-s + (−0.698 − 0.715i)25-s + (−0.365 − 0.930i)27-s + (−0.921 − 0.388i)28-s − 0.0498·31-s + (−0.698 + 0.715i)36-s + (0.890 + 0.349i)37-s + ⋯
L(s)  = 1  + (0.797 − 0.603i)3-s + (−0.878 − 0.478i)4-s + (0.995 − 0.0995i)7-s + (0.270 − 0.962i)9-s + (−0.988 + 0.149i)12-s + (0.145 + 0.201i)13-s + (0.542 + 0.840i)16-s + (0.900 − 0.433i)19-s + (0.733 − 0.680i)21-s + (−0.698 − 0.715i)25-s + (−0.365 − 0.930i)27-s + (−0.921 − 0.388i)28-s − 0.0498·31-s + (−0.698 + 0.715i)36-s + (0.890 + 0.349i)37-s + ⋯

Functional equation

Λ(s)=(2793s/2 ΓC(s) L(s)=((0.252+0.967i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2793s/2 ΓC(s) L(s)=((0.252+0.967i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 27932793    =    3⋅72⋅193 \cdot 7^{2} \cdot 19
Sign: 0.252+0.967i0.252 + 0.967i
Analytic conductor: 1.393881.39388
Root analytic conductor: 1.180631.18063
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2793(584,⋅)\chi_{2793} (584, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2793, ( :0), 0.252+0.967i)(2,\ 2793,\ (\ :0),\ 0.252 + 0.967i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.4968176281.496817628
L(12)L(\frac12) ≈\approx 1.4968176281.496817628
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1+(−0.797+0.603i)T 1 + (-0.797 + 0.603i)T
7 1+(−0.995+0.0995i)T 1 + (-0.995 + 0.0995i)T
19 1+(−0.900+0.433i)T 1 + (-0.900 + 0.433i)T
good2 1+(0.878+0.478i)T2 1 + (0.878 + 0.478i)T^{2}
5 1+(0.698+0.715i)T2 1 + (0.698 + 0.715i)T^{2}
11 1+(−0.365−0.930i)T2 1 + (-0.365 - 0.930i)T^{2}
13 1+(−0.145−0.201i)T+(−0.318+0.947i)T2 1 + (-0.145 - 0.201i)T + (-0.318 + 0.947i)T^{2}
17 1+(0.542−0.840i)T2 1 + (0.542 - 0.840i)T^{2}
23 1+(−0.456+0.889i)T2 1 + (-0.456 + 0.889i)T^{2}
29 1+(−0.797+0.603i)T2 1 + (-0.797 + 0.603i)T^{2}
31 1+0.0498T+T2 1 + 0.0498T + T^{2}
37 1+(−0.890−0.349i)T+(0.733+0.680i)T2 1 + (-0.890 - 0.349i)T + (0.733 + 0.680i)T^{2}
41 1+(−0.698−0.715i)T2 1 + (-0.698 - 0.715i)T^{2}
43 1+(0.247−0.482i)T+(−0.583−0.811i)T2 1 + (0.247 - 0.482i)T + (-0.583 - 0.811i)T^{2}
47 1+(−0.318+0.947i)T2 1 + (-0.318 + 0.947i)T^{2}
53 1+(0.542+0.840i)T2 1 + (0.542 + 0.840i)T^{2}
59 1+(−0.995+0.0995i)T2 1 + (-0.995 + 0.0995i)T^{2}
61 1+(1.89−0.636i)T+(0.797−0.603i)T2 1 + (1.89 - 0.636i)T + (0.797 - 0.603i)T^{2}
67 1+(1.27+1.52i)T+(−0.173+0.984i)T2 1 + (1.27 + 1.52i)T + (-0.173 + 0.984i)T^{2}
71 1+(0.921−0.388i)T2 1 + (0.921 - 0.388i)T^{2}
73 1+(−0.950+0.428i)T+(0.661−0.749i)T2 1 + (-0.950 + 0.428i)T + (0.661 - 0.749i)T^{2}
79 1+(−0.135−0.372i)T+(−0.766+0.642i)T2 1 + (-0.135 - 0.372i)T + (-0.766 + 0.642i)T^{2}
83 1+(−0.988+0.149i)T2 1 + (-0.988 + 0.149i)T^{2}
89 1+(−0.878+0.478i)T2 1 + (-0.878 + 0.478i)T^{2}
97 1+(−0.140+0.0511i)T+(0.766−0.642i)T2 1 + (-0.140 + 0.0511i)T + (0.766 - 0.642i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−8.866782828014789649460834369672, −7.979789484851636752599735739440, −7.68084672051546848370696664527, −6.55656191886376751339482362450, −5.77042532267771542208078961443, −4.78598847315277647855069613603, −4.16865741175530909167248554579, −3.12692273044954609262036400063, −1.94014339088109198051324721988, −1.02569329733898939482475152234, 1.53026188493785237850534419077, 2.78154784859434806254227044657, 3.65309276393886873978996082463, 4.33695780715441838503783617676, 5.09902578678749943418193988980, 5.75182960840384803649559402097, 7.35742436580405980369038368383, 7.79845754254051663150662272460, 8.411024771230598672487775277122, 9.113865886549118118564407796875

Graph of the ZZ-function along the critical line