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Properties

Label 2-3264-17.16-c1-0-57
Degree 22
Conductor 32643264
Sign −0.970+0.242i-0.970 + 0.242i
Analytic cond. 26.063126.0631
Root an. cond. 5.105215.10521
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·3-s + 2i·5-s + 2i·7-s − 9-s − 2·13-s + 2·15-s + (−1 − 4i)17-s − 4·19-s + 2·21-s + 2i·23-s + 25-s + i·27-s + 2i·29-s + 2i·31-s − 4·35-s + ⋯
L(s)  = 1  − 0.577i·3-s + 0.894i·5-s + 0.755i·7-s − 0.333·9-s − 0.554·13-s + 0.516·15-s + (−0.242 − 0.970i)17-s − 0.917·19-s + 0.436·21-s + 0.417i·23-s + 0.200·25-s + 0.192i·27-s + 0.371i·29-s + 0.359i·31-s − 0.676·35-s + ⋯

Functional equation

Λ(s)=(3264s/2 ΓC(s) L(s)=((−0.970+0.242i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3264s/2 ΓC(s+1/2) L(s)=((−0.970+0.242i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32643264    =    26⋅3⋅172^{6} \cdot 3 \cdot 17
Sign: −0.970+0.242i-0.970 + 0.242i
Analytic conductor: 26.063126.0631
Root analytic conductor: 5.105215.10521
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3264(577,⋅)\chi_{3264} (577, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 11
Selberg data: (2, 3264, ( :1/2), −0.970+0.242i)(2,\ 3264,\ (\ :1/2),\ -0.970 + 0.242i)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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
3 1+iT 1 + iT
17 1+(1+4i)T 1 + (1 + 4i)T
good5 1−2iT−5T2 1 - 2iT - 5T^{2}
7 1−2iT−7T2 1 - 2iT - 7T^{2}
11 1−11T2 1 - 11T^{2}
13 1+2T+13T2 1 + 2T + 13T^{2}
19 1+4T+19T2 1 + 4T + 19T^{2}
23 1−2iT−23T2 1 - 2iT - 23T^{2}
29 1−2iT−29T2 1 - 2iT - 29T^{2}
31 1−2iT−31T2 1 - 2iT - 31T^{2}
37 1+2iT−37T2 1 + 2iT - 37T^{2}
41 1−41T2 1 - 41T^{2}
43 1+8T+43T2 1 + 8T + 43T^{2}
47 1+47T2 1 + 47T^{2}
53 1+2T+53T2 1 + 2T + 53T^{2}
59 1+4T+59T2 1 + 4T + 59T^{2}
61 1+6iT−61T2 1 + 6iT - 61T^{2}
67 1+12T+67T2 1 + 12T + 67T^{2}
71 1−2iT−71T2 1 - 2iT - 71T^{2}
73 1−73T2 1 - 73T^{2}
79 1+10iT−79T2 1 + 10iT - 79T^{2}
83 1+12T+83T2 1 + 12T + 83T^{2}
89 1+6T+89T2 1 + 6T + 89T^{2}
97 1+4iT−97T2 1 + 4iT - 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

−8.308214659363619700251168385294, −7.34927671293210359354178302810, −6.91196014220385073134643908580, −6.16862100960228138437264922416, −5.36373660397584487843703308909, −4.50696747772029112664654524817, −3.20349456927637199665081096295, −2.63384933237430905026090025816, −1.71970836998179909189495326566, 0, 1.33236530530530222750183972939, 2.53177668592740285158076870180, 3.75195544275812733848703153186, 4.39562714406982276701382240793, 4.95037755790314960174178662396, 5.93011284460378139820141700956, 6.69344832447220363833996743583, 7.62681910073560329533321596098, 8.428602581108460320200220546494

Graph of the ZZ-function along the critical line