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Properties

Label 2-296-296.85-c1-0-30
Degree 22
Conductor 296296
Sign 0.896+0.443i0.896 + 0.443i
Analytic cond. 2.363572.36357
Root an. cond. 1.537391.53739
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  + (1.35 − 0.392i)2-s + (0.727 + 0.420i)3-s + (1.69 − 1.06i)4-s + (0.249 − 0.432i)5-s + (1.15 + 0.285i)6-s + (0.235 − 0.407i)7-s + (1.88 − 2.11i)8-s + (−1.14 − 1.98i)9-s + (0.169 − 0.685i)10-s + 1.14i·11-s + (1.67 − 0.0646i)12-s + (−3.10 + 5.37i)13-s + (0.159 − 0.646i)14-s + (0.363 − 0.209i)15-s + (1.72 − 3.60i)16-s + (−0.363 + 0.209i)17-s + ⋯
L(s)  = 1  + (0.960 − 0.277i)2-s + (0.420 + 0.242i)3-s + (0.846 − 0.532i)4-s + (0.111 − 0.193i)5-s + (0.470 + 0.116i)6-s + (0.0889 − 0.154i)7-s + (0.665 − 0.746i)8-s + (−0.382 − 0.662i)9-s + (0.0536 − 0.216i)10-s + 0.346i·11-s + (0.484 − 0.0186i)12-s + (−0.861 + 1.49i)13-s + (0.0427 − 0.172i)14-s + (0.0937 − 0.0541i)15-s + (0.431 − 0.901i)16-s + (−0.0881 + 0.0508i)17-s + ⋯

Functional equation

Λ(s)=(296s/2 ΓC(s) L(s)=((0.896+0.443i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 + 0.443i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(296s/2 ΓC(s+1/2) L(s)=((0.896+0.443i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.896 + 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 296296    =    23⋅372^{3} \cdot 37
Sign: 0.896+0.443i0.896 + 0.443i
Analytic conductor: 2.363572.36357
Root analytic conductor: 1.537391.53739
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ296(85,⋅)\chi_{296} (85, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 296, ( :1/2), 0.896+0.443i)(2,\ 296,\ (\ :1/2),\ 0.896 + 0.443i)

Particular Values

L(1)L(1) ≈\approx 2.41658−0.565431i2.41658 - 0.565431i
L(12)L(\frac12) ≈\approx 2.41658−0.565431i2.41658 - 0.565431i
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)
bad2 1+(−1.35+0.392i)T 1 + (-1.35 + 0.392i)T
37 1+(5.34−2.90i)T 1 + (5.34 - 2.90i)T
good3 1+(−0.727−0.420i)T+(1.5+2.59i)T2 1 + (-0.727 - 0.420i)T + (1.5 + 2.59i)T^{2}
5 1+(−0.249+0.432i)T+(−2.5−4.33i)T2 1 + (-0.249 + 0.432i)T + (-2.5 - 4.33i)T^{2}
7 1+(−0.235+0.407i)T+(−3.5−6.06i)T2 1 + (-0.235 + 0.407i)T + (-3.5 - 6.06i)T^{2}
11 1−1.14iT−11T2 1 - 1.14iT - 11T^{2}
13 1+(3.10−5.37i)T+(−6.5−11.2i)T2 1 + (3.10 - 5.37i)T + (-6.5 - 11.2i)T^{2}
17 1+(0.363−0.209i)T+(8.5−14.7i)T2 1 + (0.363 - 0.209i)T + (8.5 - 14.7i)T^{2}
19 1+(−1.39+2.41i)T+(−9.5−16.4i)T2 1 + (-1.39 + 2.41i)T + (-9.5 - 16.4i)T^{2}
23 1−7.74iT−23T2 1 - 7.74iT - 23T^{2}
29 1+5.48T+29T2 1 + 5.48T + 29T^{2}
31 1+7.40iT−31T2 1 + 7.40iT - 31T^{2}
41 1+(−1.51+2.61i)T+(−20.5−35.5i)T2 1 + (-1.51 + 2.61i)T + (-20.5 - 35.5i)T^{2}
43 1+1.44T+43T2 1 + 1.44T + 43T^{2}
47 1+0.412T+47T2 1 + 0.412T + 47T^{2}
53 1+(6.89−3.98i)T+(26.5−45.8i)T2 1 + (6.89 - 3.98i)T + (26.5 - 45.8i)T^{2}
59 1+(5.73+9.92i)T+(−29.5+51.0i)T2 1 + (5.73 + 9.92i)T + (-29.5 + 51.0i)T^{2}
61 1+(−2.66+4.62i)T+(−30.5−52.8i)T2 1 + (-2.66 + 4.62i)T + (-30.5 - 52.8i)T^{2}
67 1+(−8.66−5.00i)T+(33.5+58.0i)T2 1 + (-8.66 - 5.00i)T + (33.5 + 58.0i)T^{2}
71 1+(−2.85+4.94i)T+(−35.5−61.4i)T2 1 + (-2.85 + 4.94i)T + (-35.5 - 61.4i)T^{2}
73 1+5.09T+73T2 1 + 5.09T + 73T^{2}
79 1+(−5.88−3.39i)T+(39.5+68.4i)T2 1 + (-5.88 - 3.39i)T + (39.5 + 68.4i)T^{2}
83 1+(−11.4+6.63i)T+(41.5−71.8i)T2 1 + (-11.4 + 6.63i)T + (41.5 - 71.8i)T^{2}
89 1+(10.0−5.80i)T+(44.5−77.0i)T2 1 + (10.0 - 5.80i)T + (44.5 - 77.0i)T^{2}
97 1+8.36iT−97T2 1 + 8.36iT - 97T^{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

−11.71210116822765485703427522306, −11.13536081922160915747505451917, −9.571343892316556800516952036506, −9.338990995007024812066100002098, −7.57819909843212481738035429198, −6.70543104862531724108505236390, −5.46101142415887051611928176693, −4.39163985640075956692283161225, −3.35598851081208381578713843260, −1.91771258799043687497159019276, 2.35262631940459937586014086656, 3.26306550065223765541691652716, 4.88296392686743763682305005574, 5.68423050371746997785378485793, 6.91723355100902994522574777407, 7.904473999726819165837067143891, 8.577560006971596775091384494378, 10.30378912350638515725274951409, 10.91376901980250347641784266266, 12.21160550329829020031426634136

Graph of the ZZ-function along the critical line