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Properties

Label 2-416-8.5-c1-0-10
Degree 22
Conductor 416416
Sign −0.707+0.707i-0.707 + 0.707i
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  − i·3-s − 3i·5-s − 3·7-s + 2·9-s − i·13-s − 3·15-s − 7·17-s − 4i·19-s + 3i·21-s − 4·23-s − 4·25-s − 5i·27-s + 4i·29-s + 8·31-s + 9i·35-s + ⋯
L(s)  = 1  − 0.577i·3-s − 1.34i·5-s − 1.13·7-s + 0.666·9-s − 0.277i·13-s − 0.774·15-s − 1.69·17-s − 0.917i·19-s + 0.654i·21-s − 0.834·23-s − 0.800·25-s − 0.962i·27-s + 0.742i·29-s + 1.43·31-s + 1.52i·35-s + ⋯

Functional equation

Λ(s)=(416s/2 ΓC(s) L(s)=((−0.707+0.707i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(416s/2 ΓC(s+1/2) L(s)=((−0.707+0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 416416    =    25⋅132^{5} \cdot 13
Sign: −0.707+0.707i-0.707 + 0.707i
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), −0.707+0.707i)(2,\ 416,\ (\ :1/2),\ -0.707 + 0.707i)

Particular Values

L(1)L(1) ≈\approx 0.389250−0.939732i0.389250 - 0.939732i
L(12)L(\frac12) ≈\approx 0.389250−0.939732i0.389250 - 0.939732i
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+iT−3T2 1 + iT - 3T^{2}
5 1+3iT−5T2 1 + 3iT - 5T^{2}
7 1+3T+7T2 1 + 3T + 7T^{2}
11 1−11T2 1 - 11T^{2}
17 1+7T+17T2 1 + 7T + 17T^{2}
19 1+4iT−19T2 1 + 4iT - 19T^{2}
23 1+4T+23T2 1 + 4T + 23T^{2}
29 1−4iT−29T2 1 - 4iT - 29T^{2}
31 1−8T+31T2 1 - 8T + 31T^{2}
37 1+7iT−37T2 1 + 7iT - 37T^{2}
41 1−2T+41T2 1 - 2T + 41T^{2}
43 1+iT−43T2 1 + iT - 43T^{2}
47 1−7T+47T2 1 - 7T + 47T^{2}
53 1+4iT−53T2 1 + 4iT - 53T^{2}
59 1+14iT−59T2 1 + 14iT - 59T^{2}
61 1−10iT−61T2 1 - 10iT - 61T^{2}
67 1−2iT−67T2 1 - 2iT - 67T^{2}
71 1−3T+71T2 1 - 3T + 71T^{2}
73 1−14T+73T2 1 - 14T + 73T^{2}
79 1−10T+79T2 1 - 10T + 79T^{2}
83 1−14iT−83T2 1 - 14iT - 83T^{2}
89 1+89T2 1 + 89T^{2}
97 1−8T+97T2 1 - 8T + 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

−10.88697112495268797091923845999, −9.752886466224685561337253962274, −9.067145886263812370717231823172, −8.212080707638445255604324338111, −7.00977517809252333308656063861, −6.29294867830547392975854592743, −4.96686704061183388838479542014, −4.01260748526645672830388922689, −2.29119443694623589441939074338, −0.64360083432287352970178500465, 2.41091872202386334518782012665, 3.56791279260911188479866218575, 4.46974671565504062997968650541, 6.25777464968427212796214593142, 6.61711645796808404555812741294, 7.73026060178007962197194849978, 9.089775066662220953126225337654, 10.01564076271265384887889462698, 10.38826353783580439289590554285, 11.36169354213970672628326571399

Graph of the ZZ-function along the critical line