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Properties

Label 2-3264-17.16-c1-0-26
Degree 22
Conductor 32643264
Sign 0.874−0.485i0.874 - 0.485i
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·3-s + 3.60i·5-s − 4i·7-s − 9-s + 3i·11-s + 13-s + 3.60·15-s + (2 + 3.60i)17-s + 3.60·19-s − 4·21-s − 7i·23-s − 7.99·25-s + i·27-s − 7.21i·29-s + 2i·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + 1.61i·5-s − 1.51i·7-s − 0.333·9-s + 0.904i·11-s + 0.277·13-s + 0.930·15-s + (0.485 + 0.874i)17-s + 0.827·19-s − 0.872·21-s − 1.45i·23-s − 1.59·25-s + 0.192i·27-s − 1.33i·29-s + 0.359i·31-s + ⋯

Functional equation

Λ(s)=(3264s/2 ΓC(s) L(s)=((0.874−0.485i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3264s/2 ΓC(s+1/2) L(s)=((0.874−0.485i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32643264    =    26⋅3⋅172^{6} \cdot 3 \cdot 17
Sign: 0.874−0.485i0.874 - 0.485i
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: 00
Selberg data: (2, 3264, ( :1/2), 0.874−0.485i)(2,\ 3264,\ (\ :1/2),\ 0.874 - 0.485i)

Particular Values

L(1)L(1) ≈\approx 1.8036302921.803630292
L(12)L(\frac12) ≈\approx 1.8036302921.803630292
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+(−2−3.60i)T 1 + (-2 - 3.60i)T
good5 1−3.60iT−5T2 1 - 3.60iT - 5T^{2}
7 1+4iT−7T2 1 + 4iT - 7T^{2}
11 1−3iT−11T2 1 - 3iT - 11T^{2}
13 1−T+13T2 1 - T + 13T^{2}
19 1−3.60T+19T2 1 - 3.60T + 19T^{2}
23 1+7iT−23T2 1 + 7iT - 23T^{2}
29 1+7.21iT−29T2 1 + 7.21iT - 29T^{2}
31 1−2iT−31T2 1 - 2iT - 31T^{2}
37 1−7.21iT−37T2 1 - 7.21iT - 37T^{2}
41 1−10.8iT−41T2 1 - 10.8iT - 41T^{2}
43 1+3.60T+43T2 1 + 3.60T + 43T^{2}
47 1+47T2 1 + 47T^{2}
53 1−4T+53T2 1 - 4T + 53T^{2}
59 1−14.4T+59T2 1 - 14.4T + 59T^{2}
61 1−61T2 1 - 61T^{2}
67 1+67T2 1 + 67T^{2}
71 1−8iT−71T2 1 - 8iT - 71T^{2}
73 1−73T2 1 - 73T^{2}
79 1+4iT−79T2 1 + 4iT - 79T^{2}
83 1+83T2 1 + 83T^{2}
89 1−12T+89T2 1 - 12T + 89T^{2}
97 1+7.21iT−97T2 1 + 7.21iT - 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

−8.374247205712808368151156336301, −7.75968841831120881833751542036, −7.13888571941631325479851450670, −6.63312572778628798154679315134, −6.08124806272107904385991961192, −4.72864426229455109336794847624, −3.87485800099550600804218691784, −3.12124941639114764390059310227, −2.17428672331473519914695527605, −0.985162184160150530291302550418, 0.68910440224044624790330607034, 1.88864760368510675509188852278, 3.10979092486744393970620329813, 3.86275130477383479128392296149, 5.12691928151590280029262630769, 5.39240957272455196914688803441, 5.81791488746789163213882576182, 7.22738570523916595852554934127, 8.171785788427623670192591178436, 8.777233650745520825554710636498

Graph of the ZZ-function along the critical line