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Properties

Label 2-2793-2793.1466-c0-0-0
Degree 22
Conductor 27932793
Sign 0.925+0.377i0.925 + 0.377i
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  + (−0.921 − 0.388i)3-s + (0.853 − 0.521i)4-s + (−0.411 + 0.911i)7-s + (0.698 + 0.715i)9-s + (−0.988 + 0.149i)12-s + (−1.58 + 0.158i)13-s + (0.456 − 0.889i)16-s + (0.900 − 0.433i)19-s + (0.733 − 0.680i)21-s + (0.969 − 0.246i)25-s + (−0.365 − 0.930i)27-s + (0.124 + 0.992i)28-s + 1.75·31-s + (0.969 + 0.246i)36-s + (0.970 + 0.381i)37-s + ⋯
L(s)  = 1  + (−0.921 − 0.388i)3-s + (0.853 − 0.521i)4-s + (−0.411 + 0.911i)7-s + (0.698 + 0.715i)9-s + (−0.988 + 0.149i)12-s + (−1.58 + 0.158i)13-s + (0.456 − 0.889i)16-s + (0.900 − 0.433i)19-s + (0.733 − 0.680i)21-s + (0.969 − 0.246i)25-s + (−0.365 − 0.930i)27-s + (0.124 + 0.992i)28-s + 1.75·31-s + (0.969 + 0.246i)36-s + (0.970 + 0.381i)37-s + ⋯

Functional equation

Λ(s)=(2793s/2 ΓC(s) L(s)=((0.925+0.377i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2793s/2 ΓC(s) L(s)=((0.925+0.377i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 27932793    =    3⋅72⋅193 \cdot 7^{2} \cdot 19
Sign: 0.925+0.377i0.925 + 0.377i
Analytic conductor: 1.393881.39388
Root analytic conductor: 1.180631.18063
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2793(1466,⋅)\chi_{2793} (1466, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2793, ( :0), 0.925+0.377i)(2,\ 2793,\ (\ :0),\ 0.925 + 0.377i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0519370421.051937042
L(12)L(\frac12) ≈\approx 1.0519370421.051937042
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+(0.921+0.388i)T 1 + (0.921 + 0.388i)T
7 1+(0.411−0.911i)T 1 + (0.411 - 0.911i)T
19 1+(−0.900+0.433i)T 1 + (-0.900 + 0.433i)T
good2 1+(−0.853+0.521i)T2 1 + (-0.853 + 0.521i)T^{2}
5 1+(−0.969+0.246i)T2 1 + (-0.969 + 0.246i)T^{2}
11 1+(−0.365−0.930i)T2 1 + (-0.365 - 0.930i)T^{2}
13 1+(1.58−0.158i)T+(0.980−0.198i)T2 1 + (1.58 - 0.158i)T + (0.980 - 0.198i)T^{2}
17 1+(0.456+0.889i)T2 1 + (0.456 + 0.889i)T^{2}
23 1+(0.998−0.0498i)T2 1 + (0.998 - 0.0498i)T^{2}
29 1+(0.921+0.388i)T2 1 + (0.921 + 0.388i)T^{2}
31 1−1.75T+T2 1 - 1.75T + T^{2}
37 1+(−0.970−0.381i)T+(0.733+0.680i)T2 1 + (-0.970 - 0.381i)T + (0.733 + 0.680i)T^{2}
41 1+(0.969−0.246i)T2 1 + (0.969 - 0.246i)T^{2}
43 1+(−1.39+0.0696i)T+(0.995−0.0995i)T2 1 + (-1.39 + 0.0696i)T + (0.995 - 0.0995i)T^{2}
47 1+(0.980−0.198i)T2 1 + (0.980 - 0.198i)T^{2}
53 1+(0.456−0.889i)T2 1 + (0.456 - 0.889i)T^{2}
59 1+(0.411−0.911i)T2 1 + (0.411 - 0.911i)T^{2}
61 1+(0.189−0.937i)T+(−0.921−0.388i)T2 1 + (0.189 - 0.937i)T + (-0.921 - 0.388i)T^{2}
67 1+(1.18−0.209i)T+(0.939−0.342i)T2 1 + (1.18 - 0.209i)T + (0.939 - 0.342i)T^{2}
71 1+(−0.124+0.992i)T2 1 + (-0.124 + 0.992i)T^{2}
73 1+(−1.62−1.16i)T+(0.318+0.947i)T2 1 + (-1.62 - 1.16i)T + (0.318 + 0.947i)T^{2}
79 1+(−0.963+1.14i)T+(−0.173−0.984i)T2 1 + (-0.963 + 1.14i)T + (-0.173 - 0.984i)T^{2}
83 1+(−0.988+0.149i)T2 1 + (-0.988 + 0.149i)T^{2}
89 1+(0.853+0.521i)T2 1 + (0.853 + 0.521i)T^{2}
97 1+(0.114+0.0960i)T+(0.173+0.984i)T2 1 + (0.114 + 0.0960i)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

−9.125158561432769666733925478837, −7.900226896964877874900785755013, −7.24283564483915142424459455778, −6.58073359077743183897772369889, −5.97056008372167601490164906940, −5.18954046465715521443659007248, −4.61528293933584479464200052446, −2.82092545140302331847217827763, −2.36023830339938739383845337704, −0.991071005433459335605816176962, 1.00773125097396495926518628745, 2.55962350837069015070621559795, 3.42916068251288897634729050093, 4.37153878324234146325624660284, 5.10329812210740629423812278359, 6.13937545410443907870006537183, 6.74449755078463292104080797569, 7.44666148392301434192534847548, 7.911694138239223052040029498025, 9.335603100401742517174566807977

Graph of the ZZ-function along the critical line