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Properties

Label 2-2793-2793.1346-c0-0-0
Degree 22
Conductor 27932793
Sign 0.457+0.889i0.457 + 0.889i
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 + (−0.0249 − 0.999i)4-s + (0.698 − 0.715i)7-s + (0.270 − 0.962i)9-s + (0.623 + 0.781i)12-s + (0.907 − 0.0908i)13-s + (−0.998 + 0.0498i)16-s + (0.0747 + 0.997i)19-s + (−0.124 + 0.992i)21-s + (0.270 − 0.962i)25-s + (0.365 + 0.930i)27-s + (−0.733 − 0.680i)28-s + (0.661 − 1.14i)31-s + (−0.969 − 0.246i)36-s + (0.629 − 0.0949i)37-s + ⋯
L(s)  = 1  + (−0.797 + 0.603i)3-s + (−0.0249 − 0.999i)4-s + (0.698 − 0.715i)7-s + (0.270 − 0.962i)9-s + (0.623 + 0.781i)12-s + (0.907 − 0.0908i)13-s + (−0.998 + 0.0498i)16-s + (0.0747 + 0.997i)19-s + (−0.124 + 0.992i)21-s + (0.270 − 0.962i)25-s + (0.365 + 0.930i)27-s + (−0.733 − 0.680i)28-s + (0.661 − 1.14i)31-s + (−0.969 − 0.246i)36-s + (0.629 − 0.0949i)37-s + ⋯

Functional equation

Λ(s)=(2793s/2 ΓC(s) L(s)=((0.457+0.889i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.457 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2793s/2 ΓC(s) L(s)=((0.457+0.889i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.457 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 27932793    =    3⋅72⋅193 \cdot 7^{2} \cdot 19
Sign: 0.457+0.889i0.457 + 0.889i
Analytic conductor: 1.393881.39388
Root analytic conductor: 1.180631.18063
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2793(1346,⋅)\chi_{2793} (1346, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2793, ( :0), 0.457+0.889i)(2,\ 2793,\ (\ :0),\ 0.457 + 0.889i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.99973874590.9997387459
L(12)L(\frac12) ≈\approx 0.99973874590.9997387459
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+(−0.698+0.715i)T 1 + (-0.698 + 0.715i)T
19 1+(−0.0747−0.997i)T 1 + (-0.0747 - 0.997i)T
good2 1+(0.0249+0.999i)T2 1 + (0.0249 + 0.999i)T^{2}
5 1+(−0.270+0.962i)T2 1 + (-0.270 + 0.962i)T^{2}
11 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
13 1+(−0.907+0.0908i)T+(0.980−0.198i)T2 1 + (-0.907 + 0.0908i)T + (0.980 - 0.198i)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+(−0.921−0.388i)T2 1 + (-0.921 - 0.388i)T^{2}
31 1+(−0.661+1.14i)T+(−0.5−0.866i)T2 1 + (-0.661 + 1.14i)T + (-0.5 - 0.866i)T^{2}
37 1+(−0.629+0.0949i)T+(0.955−0.294i)T2 1 + (-0.629 + 0.0949i)T + (0.955 - 0.294i)T^{2}
41 1+(0.969−0.246i)T2 1 + (0.969 - 0.246i)T^{2}
43 1+(1.98−0.0991i)T+(0.995−0.0995i)T2 1 + (1.98 - 0.0991i)T + (0.995 - 0.0995i)T^{2}
47 1+(0.318−0.947i)T2 1 + (0.318 - 0.947i)T^{2}
53 1+(0.998−0.0498i)T2 1 + (0.998 - 0.0498i)T^{2}
59 1+(−0.995+0.0995i)T2 1 + (-0.995 + 0.0995i)T^{2}
61 1+(−0.421−1.25i)T+(−0.797+0.603i)T2 1 + (-0.421 - 1.25i)T + (-0.797 + 0.603i)T^{2}
67 1+(0.857+0.312i)T+(0.766+0.642i)T2 1 + (0.857 + 0.312i)T + (0.766 + 0.642i)T^{2}
71 1+(0.124−0.992i)T2 1 + (0.124 - 0.992i)T^{2}
73 1+(0.806+1.78i)T+(−0.661+0.749i)T2 1 + (0.806 + 1.78i)T + (-0.661 + 0.749i)T^{2}
79 1+(−0.305+1.72i)T+(−0.939−0.342i)T2 1 + (-0.305 + 1.72i)T + (-0.939 - 0.342i)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+(−1.26−1.06i)T+(0.173+0.984i)T2 1 + (-1.26 - 1.06i)T + (0.173 + 0.984i)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

−8.954208257229999548377751228109, −8.182363609322613870370279902890, −7.20872902241092365603312966982, −6.17253316403591295378735098556, −5.96578199609357901340766801327, −4.84273148212416096469176815187, −4.41162624280751203948951118893, −3.45524475264355129522969377385, −1.80043595287506752746670873125, −0.798792268970554864845252564369, 1.37743466715932231421529577595, 2.46188600174208019343550371977, 3.43582733237908922080763361917, 4.64863265372302125525436254570, 5.16157476158867050173275997025, 6.17810559761934592645806407984, 6.88747010663576540343950947427, 7.54592098685592540581246966490, 8.489299611652349598082946508204, 8.671995038444861525846502440164

Graph of the ZZ-function along the critical line