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Properties

Label 2-416-13.9-c1-0-8
Degree 22
Conductor 416416
Sign 0.0128+0.999i0.0128 + 0.999i
Analytic cond. 3.321773.32177
Root an. cond. 1.822571.82257
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  + (−1.65 + 2.87i)3-s + (−1.65 − 2.87i)7-s + (−4 − 6.92i)9-s + (−1.65 + 2.87i)11-s + (−1 − 3.46i)13-s + (1.5 + 2.59i)17-s + (−1.65 − 2.87i)19-s + 11·21-s + (1.65 − 2.87i)23-s − 5·25-s + 16.5·27-s + (2.5 − 4.33i)29-s + (−5.5 − 9.52i)33-s + (−4.5 + 7.79i)37-s + (11.6 + 2.87i)39-s + ⋯
L(s)  = 1  + (−0.957 + 1.65i)3-s + (−0.626 − 1.08i)7-s + (−1.33 − 2.30i)9-s + (−0.500 + 0.866i)11-s + (−0.277 − 0.960i)13-s + (0.363 + 0.630i)17-s + (−0.380 − 0.658i)19-s + 2.40·21-s + (0.345 − 0.598i)23-s − 25-s + 3.19·27-s + (0.464 − 0.804i)29-s + (−0.957 − 1.65i)33-s + (−0.739 + 1.28i)37-s + (1.85 + 0.459i)39-s + ⋯

Functional equation

Λ(s)=(416s/2 ΓC(s) L(s)=((0.0128+0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(416s/2 ΓC(s+1/2) L(s)=((0.0128+0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 416416    =    25⋅132^{5} \cdot 13
Sign: 0.0128+0.999i0.0128 + 0.999i
Analytic conductor: 3.321773.32177
Root analytic conductor: 1.822571.82257
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ416(321,⋅)\chi_{416} (321, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 416, ( :1/2), 0.0128+0.999i)(2,\ 416,\ (\ :1/2),\ 0.0128 + 0.999i)

Particular Values

L(1)L(1) ≈\approx 0.172266−0.170071i0.172266 - 0.170071i
L(12)L(\frac12) ≈\approx 0.172266−0.170071i0.172266 - 0.170071i
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
13 1+(1+3.46i)T 1 + (1 + 3.46i)T
good3 1+(1.65−2.87i)T+(−1.5−2.59i)T2 1 + (1.65 - 2.87i)T + (-1.5 - 2.59i)T^{2}
5 1+5T2 1 + 5T^{2}
7 1+(1.65+2.87i)T+(−3.5+6.06i)T2 1 + (1.65 + 2.87i)T + (-3.5 + 6.06i)T^{2}
11 1+(1.65−2.87i)T+(−5.5−9.52i)T2 1 + (1.65 - 2.87i)T + (-5.5 - 9.52i)T^{2}
17 1+(−1.5−2.59i)T+(−8.5+14.7i)T2 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2}
19 1+(1.65+2.87i)T+(−9.5+16.4i)T2 1 + (1.65 + 2.87i)T + (-9.5 + 16.4i)T^{2}
23 1+(−1.65+2.87i)T+(−11.5−19.9i)T2 1 + (-1.65 + 2.87i)T + (-11.5 - 19.9i)T^{2}
29 1+(−2.5+4.33i)T+(−14.5−25.1i)T2 1 + (-2.5 + 4.33i)T + (-14.5 - 25.1i)T^{2}
31 1+31T2 1 + 31T^{2}
37 1+(4.5−7.79i)T+(−18.5−32.0i)T2 1 + (4.5 - 7.79i)T + (-18.5 - 32.0i)T^{2}
41 1+(−1.5+2.59i)T+(−20.5−35.5i)T2 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2}
43 1+(4.97+8.61i)T+(−21.5+37.2i)T2 1 + (4.97 + 8.61i)T + (-21.5 + 37.2i)T^{2}
47 1+6.63T+47T2 1 + 6.63T + 47T^{2}
53 1+8T+53T2 1 + 8T + 53T^{2}
59 1+(1.65+2.87i)T+(−29.5+51.0i)T2 1 + (1.65 + 2.87i)T + (-29.5 + 51.0i)T^{2}
61 1+(−4.5−7.79i)T+(−30.5+52.8i)T2 1 + (-4.5 - 7.79i)T + (-30.5 + 52.8i)T^{2}
67 1+(4.97−8.61i)T+(−33.5−58.0i)T2 1 + (4.97 - 8.61i)T + (-33.5 - 58.0i)T^{2}
71 1+(4.97+8.61i)T+(−35.5+61.4i)T2 1 + (4.97 + 8.61i)T + (-35.5 + 61.4i)T^{2}
73 1+4T+73T2 1 + 4T + 73T^{2}
79 1−6.63T+79T2 1 - 6.63T + 79T^{2}
83 1+13.2T+83T2 1 + 13.2T + 83T^{2}
89 1+(−0.5+0.866i)T+(−44.5−77.0i)T2 1 + (-0.5 + 0.866i)T + (-44.5 - 77.0i)T^{2}
97 1+(3.5+6.06i)T+(−48.5+84.0i)T2 1 + (3.5 + 6.06i)T + (-48.5 + 84.0i)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

−10.59228494195776539483679486489, −10.20942033477625921288845056072, −9.715342692044473904762367653723, −8.408443863253367011453782325409, −7.05266527011519654170626685934, −6.03063996788653119022733098805, −4.98119725872838407342100120594, −4.20983021190472638747178398866, −3.18986675445017056955313795116, −0.16880055800898607543769502140, 1.72082436000843088939663263624, 2.94841899390525123404647264971, 5.14897833310567816177935343407, 5.92359059338008495314646692299, 6.59656335498201544698006380011, 7.60017885510005422526492770268, 8.463771853109622890807897440744, 9.568903761745710862371818343130, 10.94805996058471822880669610983, 11.64841941371666837721278998082

Graph of the ZZ-function along the critical line