Properties

Label 2-2793-2793.59-c0-0-0
Degree 22
Conductor 27932793
Sign 0.3200.947i0.320 - 0.947i
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  + (−0.995 − 0.0995i)3-s + (0.797 + 0.603i)4-s + (0.878 + 0.478i)7-s + (0.980 + 0.198i)9-s + (−0.733 − 0.680i)12-s + (0.0291 + 1.16i)13-s + (0.270 + 0.962i)16-s + (−0.623 + 0.781i)19-s + (−0.826 − 0.563i)21-s + (0.661 − 0.749i)25-s + (−0.955 − 0.294i)27-s + (0.411 + 0.911i)28-s − 0.248·31-s + (0.661 + 0.749i)36-s + (−0.355 − 1.15i)37-s + ⋯
L(s)  = 1  + (−0.995 − 0.0995i)3-s + (0.797 + 0.603i)4-s + (0.878 + 0.478i)7-s + (0.980 + 0.198i)9-s + (−0.733 − 0.680i)12-s + (0.0291 + 1.16i)13-s + (0.270 + 0.962i)16-s + (−0.623 + 0.781i)19-s + (−0.826 − 0.563i)21-s + (0.661 − 0.749i)25-s + (−0.955 − 0.294i)27-s + (0.411 + 0.911i)28-s − 0.248·31-s + (0.661 + 0.749i)36-s + (−0.355 − 1.15i)37-s + ⋯

Functional equation

Λ(s)=(2793s/2ΓC(s)L(s)=((0.3200.947i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.320 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2793s/2ΓC(s)L(s)=((0.3200.947i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.320 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 27932793    =    372193 \cdot 7^{2} \cdot 19
Sign: 0.3200.947i0.320 - 0.947i
Analytic conductor: 1.393881.39388
Root analytic conductor: 1.180631.18063
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2793(59,)\chi_{2793} (59, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2793, ( :0), 0.3200.947i)(2,\ 2793,\ (\ :0),\ 0.320 - 0.947i)

Particular Values

L(12)L(\frac{1}{2}) \approx 1.1681275001.168127500
L(12)L(\frac12) \approx 1.1681275001.168127500
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1+(0.995+0.0995i)T 1 + (0.995 + 0.0995i)T
7 1+(0.8780.478i)T 1 + (-0.878 - 0.478i)T
19 1+(0.6230.781i)T 1 + (0.623 - 0.781i)T
good2 1+(0.7970.603i)T2 1 + (-0.797 - 0.603i)T^{2}
5 1+(0.661+0.749i)T2 1 + (-0.661 + 0.749i)T^{2}
11 1+(0.9550.294i)T2 1 + (-0.955 - 0.294i)T^{2}
13 1+(0.02911.16i)T+(0.998+0.0498i)T2 1 + (-0.0291 - 1.16i)T + (-0.998 + 0.0498i)T^{2}
17 1+(0.2700.962i)T2 1 + (0.270 - 0.962i)T^{2}
23 1+(0.698+0.715i)T2 1 + (-0.698 + 0.715i)T^{2}
29 1+(0.995+0.0995i)T2 1 + (0.995 + 0.0995i)T^{2}
31 1+0.248T+T2 1 + 0.248T + T^{2}
37 1+(0.355+1.15i)T+(0.826+0.563i)T2 1 + (0.355 + 1.15i)T + (-0.826 + 0.563i)T^{2}
41 1+(0.6610.749i)T2 1 + (0.661 - 0.749i)T^{2}
43 1+(1.361.40i)T+(0.02490.999i)T2 1 + (1.36 - 1.40i)T + (-0.0249 - 0.999i)T^{2}
47 1+(0.998+0.0498i)T2 1 + (-0.998 + 0.0498i)T^{2}
53 1+(0.270+0.962i)T2 1 + (0.270 + 0.962i)T^{2}
59 1+(0.8780.478i)T2 1 + (-0.878 - 0.478i)T^{2}
61 1+(0.0989+1.98i)T+(0.9950.0995i)T2 1 + (-0.0989 + 1.98i)T + (-0.995 - 0.0995i)T^{2}
67 1+(0.555+1.52i)T+(0.7660.642i)T2 1 + (-0.555 + 1.52i)T + (-0.766 - 0.642i)T^{2}
71 1+(0.411+0.911i)T2 1 + (-0.411 + 0.911i)T^{2}
73 1+(0.4050.662i)T+(0.4560.889i)T2 1 + (0.405 - 0.662i)T + (-0.456 - 0.889i)T^{2}
79 1+(1.650.291i)T+(0.939+0.342i)T2 1 + (-1.65 - 0.291i)T + (0.939 + 0.342i)T^{2}
83 1+(0.7330.680i)T2 1 + (-0.733 - 0.680i)T^{2}
89 1+(0.7970.603i)T2 1 + (0.797 - 0.603i)T^{2}
97 1+(0.1260.719i)T+(0.9390.342i)T2 1 + (0.126 - 0.719i)T + (-0.939 - 0.342i)T^{2}
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   L(s)=p j=12(1αj,pps)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

−9.052942964092469222530198109160, −8.180479836816819405950066189998, −7.63440060872504004188407868388, −6.60127022981642426696479412788, −6.36173208351702387120660740339, −5.28457978908542921449131386640, −4.52133199842339210066712441502, −3.66798667894789175671536885976, −2.24111002940490541059634356186, −1.60367011562447023104986250902, 0.876152224907721286852884589432, 1.84755600743347367936142450609, 3.11877183712131143467764878037, 4.33644953942454780574853568770, 5.23053078649997077764970330092, 5.54974581311665881689706655915, 6.67198288578743062934858346463, 7.07006309192422271934809489011, 7.905029934748291642530194987633, 8.824791817086813376618559023465

Graph of the ZZ-function along the critical line