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Properties

Label 2-253-253.87-c2-0-22
Degree 22
Conductor 253253
Sign 0.643−0.765i0.643 - 0.765i
Analytic cond. 6.893756.89375
Root an. cond. 2.625592.62559
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.615 + 0.710i)3-s + (3.36 + 2.16i)4-s + (0.262 − 1.82i)5-s + (1.15 + 8.03i)9-s + (10.5 − 3.09i)11-s + (−3.60 + 1.05i)12-s + (1.13 + 1.30i)15-s + (6.64 + 14.5i)16-s + (4.82 − 5.57i)20-s + (−12.5 + 19.2i)23-s + (20.7 + 6.08i)25-s + (−13.5 − 8.69i)27-s + (28.7 + 33.2i)31-s + (−4.29 + 9.40i)33-s + (−13.4 + 29.5i)36-s + (−2.36 − 16.4i)37-s + ⋯
L(s)  = 1  + (−0.205 + 0.236i)3-s + (0.841 + 0.540i)4-s + (0.0524 − 0.364i)5-s + (0.128 + 0.892i)9-s + (0.959 − 0.281i)11-s + (−0.300 + 0.0882i)12-s + (0.0756 + 0.0873i)15-s + (0.415 + 0.909i)16-s + (0.241 − 0.278i)20-s + (−0.547 + 0.836i)23-s + (0.829 + 0.243i)25-s + (−0.501 − 0.322i)27-s + (0.929 + 1.07i)31-s + (−0.130 + 0.284i)33-s + (−0.374 + 0.820i)36-s + (−0.0639 − 0.444i)37-s + ⋯

Functional equation

Λ(s)=(253s/2 ΓC(s) L(s)=((0.643−0.765i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.643 - 0.765i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(253s/2 ΓC(s+1) L(s)=((0.643−0.765i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.643 - 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 253253    =    11⋅2311 \cdot 23
Sign: 0.643−0.765i0.643 - 0.765i
Analytic conductor: 6.893756.89375
Root analytic conductor: 2.625592.62559
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ253(87,⋅)\chi_{253} (87, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 253, ( :1), 0.643−0.765i)(2,\ 253,\ (\ :1),\ 0.643 - 0.765i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.69570+0.790048i1.69570 + 0.790048i
L(12)L(\frac12) ≈\approx 1.69570+0.790048i1.69570 + 0.790048i
L(2)L(2) 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)
bad11 1+(−10.5+3.09i)T 1 + (-10.5 + 3.09i)T
23 1+(12.5−19.2i)T 1 + (12.5 - 19.2i)T
good2 1+(−3.36−2.16i)T2 1 + (-3.36 - 2.16i)T^{2}
3 1+(0.615−0.710i)T+(−1.28−8.90i)T2 1 + (0.615 - 0.710i)T + (-1.28 - 8.90i)T^{2}
5 1+(−0.262+1.82i)T+(−23.9−7.04i)T2 1 + (-0.262 + 1.82i)T + (-23.9 - 7.04i)T^{2}
7 1+(32.0−37.0i)T2 1 + (32.0 - 37.0i)T^{2}
13 1+(110.+127.i)T2 1 + (110. + 127. i)T^{2}
17 1+(−120.+262.i)T2 1 + (-120. + 262. i)T^{2}
19 1+(−149.−328.i)T2 1 + (-149. - 328. i)T^{2}
29 1+(−349.+765.i)T2 1 + (-349. + 765. i)T^{2}
31 1+(−28.7−33.2i)T+(−136.+951.i)T2 1 + (-28.7 - 33.2i)T + (-136. + 951. i)T^{2}
37 1+(2.36+16.4i)T+(−1.31e3+385.i)T2 1 + (2.36 + 16.4i)T + (-1.31e3 + 385. i)T^{2}
41 1+(1.61e3+473.i)T2 1 + (1.61e3 + 473. i)T^{2}
43 1+(263.+1.83e3i)T2 1 + (263. + 1.83e3i)T^{2}
47 1−71.6T+2.20e3T2 1 - 71.6T + 2.20e3T^{2}
53 1+(42.3+92.7i)T+(−1.83e3+2.12e3i)T2 1 + (42.3 + 92.7i)T + (-1.83e3 + 2.12e3i)T^{2}
59 1+(−48.5+106.i)T+(−2.27e3−2.63e3i)T2 1 + (-48.5 + 106. i)T + (-2.27e3 - 2.63e3i)T^{2}
61 1+(529.−3.68e3i)T2 1 + (529. - 3.68e3i)T^{2}
67 1+(126.+37.2i)T+(3.77e3+2.42e3i)T2 1 + (126. + 37.2i)T + (3.77e3 + 2.42e3i)T^{2}
71 1+(135.+39.9i)T+(4.24e3+2.72e3i)T2 1 + (135. + 39.9i)T + (4.24e3 + 2.72e3i)T^{2}
73 1+(−2.21e3−4.84e3i)T2 1 + (-2.21e3 - 4.84e3i)T^{2}
79 1+(4.08e3+4.71e3i)T2 1 + (4.08e3 + 4.71e3i)T^{2}
83 1+(6.60e3−1.94e3i)T2 1 + (6.60e3 - 1.94e3i)T^{2}
89 1+(−115.+133.i)T+(−1.12e3−7.84e3i)T2 1 + (-115. + 133. i)T + (-1.12e3 - 7.84e3i)T^{2}
97 1+(−25.7+179.i)T+(−9.02e3−2.65e3i)T2 1 + (-25.7 + 179. i)T + (-9.02e3 - 2.65e3i)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

−11.84123713957151881566632290730, −11.11139263090727559578810670092, −10.21055114877528576547865775102, −8.964139267502373102317281823453, −7.989293182771418773714737513786, −6.99572267936048805593015819116, −5.91090158079805622357146906748, −4.61390226861914141717539427941, −3.28666650943097015643928597924, −1.70128506675823800958693068423, 1.13105594665935182124037643053, 2.69791902219236920881109064282, 4.25472863532645696695728870452, 5.94622930451951237042750794503, 6.55413675259822337143698782971, 7.40230334578706573414250855673, 8.900148489054212052139254658729, 9.924784880537566336482698130260, 10.73637113258170331633975171190, 11.86236289528458650087528253797

Graph of the ZZ-function along the critical line