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Properties

Label 2-416-8.5-c1-0-9
Degree 22
Conductor 416416
Sign −0.258+0.965i-0.258 + 0.965i
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  − 2i·3-s − 3.46i·5-s + 4.73·7-s − 9-s + 1.26i·11-s + i·13-s − 6.92·15-s − 1.46·17-s + 2.73i·19-s − 9.46i·21-s − 4·23-s − 6.99·25-s − 4i·27-s + 2i·29-s + 3.26·31-s + ⋯
L(s)  = 1  − 1.15i·3-s − 1.54i·5-s + 1.78·7-s − 0.333·9-s + 0.382i·11-s + 0.277i·13-s − 1.78·15-s − 0.355·17-s + 0.626i·19-s − 2.06i·21-s − 0.834·23-s − 1.39·25-s − 0.769i·27-s + 0.371i·29-s + 0.586·31-s + ⋯

Functional equation

Λ(s)=(416s/2 ΓC(s) L(s)=((−0.258+0.965i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(416s/2 ΓC(s+1/2) L(s)=((−0.258+0.965i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 416416    =    25⋅132^{5} \cdot 13
Sign: −0.258+0.965i-0.258 + 0.965i
Analytic conductor: 3.321773.32177
Root analytic conductor: 1.822571.82257
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ416(209,⋅)\chi_{416} (209, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 416, ( :1/2), −0.258+0.965i)(2,\ 416,\ (\ :1/2),\ -0.258 + 0.965i)

Particular Values

L(1)L(1) ≈\approx 0.987792−1.28731i0.987792 - 1.28731i
L(12)L(\frac12) ≈\approx 0.987792−1.28731i0.987792 - 1.28731i
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−iT 1 - iT
good3 1+2iT−3T2 1 + 2iT - 3T^{2}
5 1+3.46iT−5T2 1 + 3.46iT - 5T^{2}
7 1−4.73T+7T2 1 - 4.73T + 7T^{2}
11 1−1.26iT−11T2 1 - 1.26iT - 11T^{2}
17 1+1.46T+17T2 1 + 1.46T + 17T^{2}
19 1−2.73iT−19T2 1 - 2.73iT - 19T^{2}
23 1+4T+23T2 1 + 4T + 23T^{2}
29 1−2iT−29T2 1 - 2iT - 29T^{2}
31 1−3.26T+31T2 1 - 3.26T + 31T^{2}
37 1−4.92iT−37T2 1 - 4.92iT - 37T^{2}
41 1+4.92T+41T2 1 + 4.92T + 41T^{2}
43 1−7.46iT−43T2 1 - 7.46iT - 43T^{2}
47 1+3.26T+47T2 1 + 3.26T + 47T^{2}
53 1−10.9iT−53T2 1 - 10.9iT - 53T^{2}
59 1+0.196iT−59T2 1 + 0.196iT - 59T^{2}
61 1+10.9iT−61T2 1 + 10.9iT - 61T^{2}
67 1−2.73iT−67T2 1 - 2.73iT - 67T^{2}
71 1+2.19T+71T2 1 + 2.19T + 71T^{2}
73 1+0.535T+73T2 1 + 0.535T + 73T^{2}
79 1−1.46T+79T2 1 - 1.46T + 79T^{2}
83 1+6.73iT−83T2 1 + 6.73iT - 83T^{2}
89 1−17.3T+89T2 1 - 17.3T + 89T^{2}
97 1+14.3T+97T2 1 + 14.3T + 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

−11.31612240228864407876460987867, −9.972408185142455694800867205414, −8.744082584547826445155678671743, −8.131188689436848486045260496386, −7.53917559313879621791417810081, −6.19942722935809631447847378801, −4.98462993962920393861146330572, −4.38201926300298958803600849563, −1.91365119986622245651772896758, −1.25787348341261807773421410633, 2.22449239575256709356198648105, 3.57585239248970004918043916706, 4.56341897588010696309581687180, 5.55997228536087349195201798109, 6.86174036002332583599412977952, 7.83355508684672240742556567360, 8.758831745200813004232375446695, 9.998905142741516134238077626976, 10.64073027091680827405028239155, 11.18347939226068674769171128578

Graph of the ZZ-function along the critical line