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Properties

Label 2-2793-2793.1871-c0-0-0
Degree 22
Conductor 27932793
Sign 0.151−0.988i0.151 - 0.988i
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  + (−0.853 + 0.521i)3-s + (−0.411 − 0.911i)4-s + (0.921 + 0.388i)7-s + (0.456 − 0.889i)9-s + (0.826 + 0.563i)12-s + (−1.40 + 1.06i)13-s + (−0.661 + 0.749i)16-s + (−0.222 + 0.974i)19-s + (−0.988 + 0.149i)21-s + (−0.998 + 0.0498i)25-s + (0.0747 + 0.997i)27-s + (−0.0249 − 0.999i)28-s + 1.99·31-s + (−0.998 − 0.0498i)36-s + (−0.0614 + 0.820i)37-s + ⋯
L(s)  = 1  + (−0.853 + 0.521i)3-s + (−0.411 − 0.911i)4-s + (0.921 + 0.388i)7-s + (0.456 − 0.889i)9-s + (0.826 + 0.563i)12-s + (−1.40 + 1.06i)13-s + (−0.661 + 0.749i)16-s + (−0.222 + 0.974i)19-s + (−0.988 + 0.149i)21-s + (−0.998 + 0.0498i)25-s + (0.0747 + 0.997i)27-s + (−0.0249 − 0.999i)28-s + 1.99·31-s + (−0.998 − 0.0498i)36-s + (−0.0614 + 0.820i)37-s + ⋯

Functional equation

Λ(s)=(2793s/2 ΓC(s) L(s)=((0.151−0.988i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2793s/2 ΓC(s) L(s)=((0.151−0.988i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 27932793    =    3⋅72⋅193 \cdot 7^{2} \cdot 19
Sign: 0.151−0.988i0.151 - 0.988i
Analytic conductor: 1.393881.39388
Root analytic conductor: 1.180631.18063
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2793(1871,⋅)\chi_{2793} (1871, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2793, ( :0), 0.151−0.988i)(2,\ 2793,\ (\ :0),\ 0.151 - 0.988i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.66373333960.6637333396
L(12)L(\frac12) ≈\approx 0.66373333960.6637333396
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+(0.853−0.521i)T 1 + (0.853 - 0.521i)T
7 1+(−0.921−0.388i)T 1 + (-0.921 - 0.388i)T
19 1+(0.222−0.974i)T 1 + (0.222 - 0.974i)T
good2 1+(0.411+0.911i)T2 1 + (0.411 + 0.911i)T^{2}
5 1+(0.998−0.0498i)T2 1 + (0.998 - 0.0498i)T^{2}
11 1+(−0.0747−0.997i)T2 1 + (-0.0747 - 0.997i)T^{2}
13 1+(1.40−1.06i)T+(0.270−0.962i)T2 1 + (1.40 - 1.06i)T + (0.270 - 0.962i)T^{2}
17 1+(0.661+0.749i)T2 1 + (0.661 + 0.749i)T^{2}
23 1+(0.318+0.947i)T2 1 + (0.318 + 0.947i)T^{2}
29 1+(0.853−0.521i)T2 1 + (0.853 - 0.521i)T^{2}
31 1−1.99T+T2 1 - 1.99T + T^{2}
37 1+(0.0614−0.820i)T+(−0.988−0.149i)T2 1 + (0.0614 - 0.820i)T + (-0.988 - 0.149i)T^{2}
41 1+(0.998−0.0498i)T2 1 + (0.998 - 0.0498i)T^{2}
43 1+(0.290+0.864i)T+(−0.797+0.603i)T2 1 + (0.290 + 0.864i)T + (-0.797 + 0.603i)T^{2}
47 1+(−0.270+0.962i)T2 1 + (-0.270 + 0.962i)T^{2}
53 1+(0.661−0.749i)T2 1 + (0.661 - 0.749i)T^{2}
59 1+(−0.921−0.388i)T2 1 + (-0.921 - 0.388i)T^{2}
61 1+(−0.538−1.91i)T+(−0.853+0.521i)T2 1 + (-0.538 - 1.91i)T + (-0.853 + 0.521i)T^{2}
67 1+(1.65−0.600i)T+(0.766−0.642i)T2 1 + (1.65 - 0.600i)T + (0.766 - 0.642i)T^{2}
71 1+(0.0249−0.999i)T2 1 + (0.0249 - 0.999i)T^{2}
73 1+(−0.145+1.15i)T+(−0.969−0.246i)T2 1 + (-0.145 + 1.15i)T + (-0.969 - 0.246i)T^{2}
79 1+(−0.242−1.37i)T+(−0.939+0.342i)T2 1 + (-0.242 - 1.37i)T + (-0.939 + 0.342i)T^{2}
83 1+(−0.826−0.563i)T2 1 + (-0.826 - 0.563i)T^{2}
89 1+(0.411−0.911i)T2 1 + (0.411 - 0.911i)T^{2}
97 1+(−0.331−1.88i)T+(−0.939+0.342i)T2 1 + (-0.331 - 1.88i)T + (-0.939 + 0.342i)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

−9.301894574867595520952729758009, −8.595767737185467204881296926874, −7.61569375761722839196269698668, −6.63630715170682483813468291878, −5.96040066318992438039005604538, −5.19763152052003472455538525086, −4.62561772435136773494467045156, −4.03217696881711834384166891841, −2.33086654544292113190713202791, −1.31447572587706667970429273326, 0.50788688657082750301010326352, 2.09262655952713429780981462467, 3.03341470744638785089300268938, 4.46755945838327533529780667199, 4.74360400980910268846322355753, 5.62982701549187073091657915331, 6.70468997813687231081478024540, 7.48043094337997840280337811723, 7.86270164913274988029060994616, 8.520869004771327215033798920723

Graph of the ZZ-function along the critical line