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Properties

Label 2-416-13.3-c1-0-5
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(289,⋅)\chi_{416} (289, \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 1.36400+1.34662i1.36400 + 1.34662i
L(12)L(\frac12) ≈\approx 1.36400+1.34662i1.36400 + 1.34662i
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

−11.08393222630473618117997142836, −10.41146494491583582180983605036, −9.555010468223417486345730071864, −9.035589203665678882415639145963, −7.87273121172963451066923323974, −7.05029437252841768071911368005, −5.15945733075600577077676271597, −4.38431073096514722446644601749, −3.73067982323631693501141951886, −2.21698153858537655787529738817, 1.31218675849413734944107603764, 2.48349446992287228183830790898, 3.53363272033310291419500223737, 5.66840111213850985847655353908, 6.20630909665338915881201577156, 7.60955100485619283118037922508, 8.137168737027138719099948848284, 8.760104854956895070987330381274, 9.834054597293179805332052482999, 11.43994153228692479353828043939

Graph of the ZZ-function along the critical line