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Properties

Label 2-261-261.101-c1-0-20
Degree 22
Conductor 261261
Sign −0.0856+0.996i-0.0856 + 0.996i
Analytic cond. 2.084092.08409
Root an. cond. 1.443631.44363
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.694 + 1.59i)2-s + (0.725 − 1.57i)3-s + (−0.690 − 0.744i)4-s + (−3.95 − 0.595i)5-s + (1.99 + 2.24i)6-s + (−0.402 − 0.373i)7-s + (−1.61 + 0.564i)8-s + (−1.94 − 2.28i)9-s + (3.69 − 5.87i)10-s + (2.35 − 2.02i)11-s + (−1.67 + 0.546i)12-s + (0.226 − 0.332i)13-s + (0.873 − 0.381i)14-s + (−3.80 + 5.78i)15-s + (0.373 − 4.98i)16-s + (−2.69 − 2.69i)17-s + ⋯
L(s)  = 1  + (−0.491 + 1.12i)2-s + (0.418 − 0.908i)3-s + (−0.345 − 0.372i)4-s + (−1.76 − 0.266i)5-s + (0.816 + 0.917i)6-s + (−0.152 − 0.141i)7-s + (−0.570 + 0.199i)8-s + (−0.648 − 0.760i)9-s + (1.16 − 1.85i)10-s + (0.709 − 0.610i)11-s + (−0.482 + 0.157i)12-s + (0.0628 − 0.0921i)13-s + (0.233 − 0.101i)14-s + (−0.982 + 1.49i)15-s + (0.0934 − 1.24i)16-s + (−0.653 − 0.653i)17-s + ⋯

Functional equation

Λ(s)=(261s/2 ΓC(s) L(s)=((−0.0856+0.996i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0856 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(261s/2 ΓC(s+1/2) L(s)=((−0.0856+0.996i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0856 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 261261    =    32⋅293^{2} \cdot 29
Sign: −0.0856+0.996i-0.0856 + 0.996i
Analytic conductor: 2.084092.08409
Root analytic conductor: 1.443631.44363
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ261(101,⋅)\chi_{261} (101, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 261, ( :1/2), −0.0856+0.996i)(2,\ 261,\ (\ :1/2),\ -0.0856 + 0.996i)

Particular Values

L(1)L(1) ≈\approx 0.266426−0.290325i0.266426 - 0.290325i
L(12)L(\frac12) ≈\approx 0.266426−0.290325i0.266426 - 0.290325i
L(32)L(\frac{3}{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)
bad3 1+(−0.725+1.57i)T 1 + (-0.725 + 1.57i)T
29 1+(−4.01−3.59i)T 1 + (-4.01 - 3.59i)T
good2 1+(0.694−1.59i)T+(−1.36−1.46i)T2 1 + (0.694 - 1.59i)T + (-1.36 - 1.46i)T^{2}
5 1+(3.95+0.595i)T+(4.77+1.47i)T2 1 + (3.95 + 0.595i)T + (4.77 + 1.47i)T^{2}
7 1+(0.402+0.373i)T+(0.523+6.98i)T2 1 + (0.402 + 0.373i)T + (0.523 + 6.98i)T^{2}
11 1+(−2.35+2.02i)T+(1.63−10.8i)T2 1 + (-2.35 + 2.02i)T + (1.63 - 10.8i)T^{2}
13 1+(−0.226+0.332i)T+(−4.74−12.1i)T2 1 + (-0.226 + 0.332i)T + (-4.74 - 12.1i)T^{2}
17 1+(2.69+2.69i)T+17iT2 1 + (2.69 + 2.69i)T + 17iT^{2}
19 1+(5.36+3.37i)T+(8.24+17.1i)T2 1 + (5.36 + 3.37i)T + (8.24 + 17.1i)T^{2}
23 1+(6.37+2.50i)T+(16.8+15.6i)T2 1 + (6.37 + 2.50i)T + (16.8 + 15.6i)T^{2}
31 1+(1.39−1.03i)T+(9.13−29.6i)T2 1 + (1.39 - 1.03i)T + (9.13 - 29.6i)T^{2}
37 1+(−2.74−7.85i)T+(−28.9+23.0i)T2 1 + (-2.74 - 7.85i)T + (-28.9 + 23.0i)T^{2}
41 1+(−3.40−0.912i)T+(35.5+20.5i)T2 1 + (-3.40 - 0.912i)T + (35.5 + 20.5i)T^{2}
43 1+(−0.908+1.23i)T+(−12.6−41.0i)T2 1 + (-0.908 + 1.23i)T + (-12.6 - 41.0i)T^{2}
47 1+(−6.35−7.38i)T+(−7.00+46.4i)T2 1 + (-6.35 - 7.38i)T + (-7.00 + 46.4i)T^{2}
53 1+(2.20+1.75i)T+(11.7+51.6i)T2 1 + (2.20 + 1.75i)T + (11.7 + 51.6i)T^{2}
59 1+(1.88−1.09i)T+(29.5−51.0i)T2 1 + (1.88 - 1.09i)T + (29.5 - 51.0i)T^{2}
61 1+(7.36−0.275i)T+(60.8−4.55i)T2 1 + (7.36 - 0.275i)T + (60.8 - 4.55i)T^{2}
67 1+(8.04−0.603i)T+(66.2−9.98i)T2 1 + (8.04 - 0.603i)T + (66.2 - 9.98i)T^{2}
71 1+(−12.9+6.25i)T+(44.2−55.5i)T2 1 + (-12.9 + 6.25i)T + (44.2 - 55.5i)T^{2}
73 1+(0.0593−0.527i)T+(−71.1−16.2i)T2 1 + (0.0593 - 0.527i)T + (-71.1 - 16.2i)T^{2}
79 1+(−2.22+11.7i)T+(−73.5−28.8i)T2 1 + (-2.22 + 11.7i)T + (-73.5 - 28.8i)T^{2}
83 1+(−2.07+6.72i)T+(−68.5−46.7i)T2 1 + (-2.07 + 6.72i)T + (-68.5 - 46.7i)T^{2}
89 1+(−0.918+0.103i)T+(86.7−19.8i)T2 1 + (-0.918 + 0.103i)T + (86.7 - 19.8i)T^{2}
97 1+(4.29+8.13i)T+(−54.6+80.1i)T2 1 + (4.29 + 8.13i)T + (-54.6 + 80.1i)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

−11.89535094820344637810833758178, −11.05235503601554990217334170070, −9.077412565334194784973465050425, −8.489216079680957556297958273096, −7.84328044663720006243926280148, −6.93548502962296026989136765372, −6.28434252899668927282753776297, −4.40453838626543848148828003975, −3.04620852526210515967118712387, −0.32870273584036611682737621257, 2.34391133563524547818684990345, 3.92978975131710892315575275533, 4.05522442140341280965279754883, 6.26029721136741163321817982085, 7.79563812957666366630074264893, 8.610077979757417550996397485964, 9.512152935482890612103541205632, 10.54760725461709287792708703012, 11.09262931089279076032473843572, 11.99246168002504904759070857389

Graph of the ZZ-function along the critical line