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Properties

Label 2-253-253.54-c2-0-7
Degree 22
Conductor 253253
Sign 0.890−0.455i0.890 - 0.455i
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  + (−2.49 − 5.45i)3-s + (−3.83 − 1.12i)4-s + (6.35 + 7.33i)5-s + (−17.6 + 20.3i)9-s + (1.56 + 10.8i)11-s + (3.41 + 23.7i)12-s + (24.1 − 52.9i)15-s + (13.4 + 8.65i)16-s + (−16.1 − 35.3i)20-s + (12.2 − 19.4i)23-s + (−9.84 + 68.5i)25-s + (103. + 30.4i)27-s + (−5.55 + 12.1i)31-s + (55.5 − 35.6i)33-s + (90.8 − 58.3i)36-s + (−28.5 + 32.9i)37-s + ⋯
L(s)  = 1  + (−0.830 − 1.81i)3-s + (−0.959 − 0.281i)4-s + (1.27 + 1.46i)5-s + (−1.96 + 2.26i)9-s + (0.142 + 0.989i)11-s + (0.284 + 1.97i)12-s + (1.61 − 3.53i)15-s + (0.841 + 0.540i)16-s + (−0.806 − 1.76i)20-s + (0.534 − 0.845i)23-s + (−0.393 + 2.74i)25-s + (3.83 + 1.12i)27-s + (−0.179 + 0.392i)31-s + (1.68 − 1.08i)33-s + (2.52 − 1.62i)36-s + (−0.771 + 0.890i)37-s + ⋯

Functional equation

Λ(s)=(253s/2 ΓC(s) L(s)=((0.890−0.455i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(253s/2 ΓC(s+1) L(s)=((0.890−0.455i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 253253    =    11⋅2311 \cdot 23
Sign: 0.890−0.455i0.890 - 0.455i
Analytic conductor: 6.893756.89375
Root analytic conductor: 2.625592.62559
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ253(54,⋅)\chi_{253} (54, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 253, ( :1), 0.890−0.455i)(2,\ 253,\ (\ :1),\ 0.890 - 0.455i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.889330+0.214351i0.889330 + 0.214351i
L(12)L(\frac12) ≈\approx 0.889330+0.214351i0.889330 + 0.214351i
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+(−1.56−10.8i)T 1 + (-1.56 - 10.8i)T
23 1+(−12.2+19.4i)T 1 + (-12.2 + 19.4i)T
good2 1+(3.83+1.12i)T2 1 + (3.83 + 1.12i)T^{2}
3 1+(2.49+5.45i)T+(−5.89+6.80i)T2 1 + (2.49 + 5.45i)T + (-5.89 + 6.80i)T^{2}
5 1+(−6.35−7.33i)T+(−3.55+24.7i)T2 1 + (-6.35 - 7.33i)T + (-3.55 + 24.7i)T^{2}
7 1+(−20.3−44.5i)T2 1 + (-20.3 - 44.5i)T^{2}
13 1+(−70.2+153.i)T2 1 + (-70.2 + 153. i)T^{2}
17 1+(−243.+156.i)T2 1 + (-243. + 156. i)T^{2}
19 1+(−303.−195.i)T2 1 + (-303. - 195. i)T^{2}
29 1+(−707.+454.i)T2 1 + (-707. + 454. i)T^{2}
31 1+(5.55−12.1i)T+(−629.−726.i)T2 1 + (5.55 - 12.1i)T + (-629. - 726. i)T^{2}
37 1+(28.5−32.9i)T+(−194.−1.35e3i)T2 1 + (28.5 - 32.9i)T + (-194. - 1.35e3i)T^{2}
41 1+(239.−1.66e3i)T2 1 + (239. - 1.66e3i)T^{2}
43 1+(1.21e3−1.39e3i)T2 1 + (1.21e3 - 1.39e3i)T^{2}
47 1−27.4T+2.20e3T2 1 - 27.4T + 2.20e3T^{2}
53 1+(−37.6−24.1i)T+(1.16e3+2.55e3i)T2 1 + (-37.6 - 24.1i)T + (1.16e3 + 2.55e3i)T^{2}
59 1+(74.5−47.9i)T+(1.44e3−3.16e3i)T2 1 + (74.5 - 47.9i)T + (1.44e3 - 3.16e3i)T^{2}
61 1+(2.43e3+2.81e3i)T2 1 + (2.43e3 + 2.81e3i)T^{2}
67 1+(−5.76+40.0i)T+(−4.30e3−1.26e3i)T2 1 + (-5.76 + 40.0i)T + (-4.30e3 - 1.26e3i)T^{2}
71 1+(9.70−67.4i)T+(−4.83e3−1.42e3i)T2 1 + (9.70 - 67.4i)T + (-4.83e3 - 1.42e3i)T^{2}
73 1+(−4.48e3−2.88e3i)T2 1 + (-4.48e3 - 2.88e3i)T^{2}
79 1+(−2.59e3+5.67e3i)T2 1 + (-2.59e3 + 5.67e3i)T^{2}
83 1+(980.+6.81e3i)T2 1 + (980. + 6.81e3i)T^{2}
89 1+(0.378+0.829i)T+(−5.18e3+5.98e3i)T2 1 + (0.378 + 0.829i)T + (-5.18e3 + 5.98e3i)T^{2}
97 1+(−124.−143.i)T+(−1.33e3+9.31e3i)T2 1 + (-124. - 143. i)T + (-1.33e3 + 9.31e3i)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

−12.09609182372278753980835649125, −10.83104113729997099749086627264, −10.26260674647533800796335668481, −8.974266715212764216582477185399, −7.55714318725869813465983475055, −6.75337229530578919642560152938, −6.04332305333306841792924048442, −5.09293235053050385985955112254, −2.63247617353325735809068674864, −1.46410750649385793680854809922, 0.58104167670097708620732072976, 3.56959985545802336590316908732, 4.62968317598652582468627007556, 5.38943834432353978658108001967, 5.92173158585242542588529248374, 8.579881530381908167951327841805, 9.059906387921310094858419963216, 9.681018079801420220752082694742, 10.50123393497857305232614441627, 11.68191991618183730473953099905

Graph of the ZZ-function along the critical line