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Properties

Label 2-2366-13.12-c1-0-62
Degree 22
Conductor 23662366
Sign 0.277+0.960i0.277 + 0.960i
Analytic cond. 18.892618.8926
Root an. cond. 4.346564.34656
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·2-s + 3.34·3-s − 4-s − 1.56i·5-s − 3.34i·6-s + i·7-s + i·8-s + 8.18·9-s − 1.56·10-s − 2.86i·11-s − 3.34·12-s + 14-s − 5.22i·15-s + 16-s + 2.22·17-s − 8.18i·18-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.93·3-s − 0.5·4-s − 0.699i·5-s − 1.36i·6-s + 0.377i·7-s + 0.353i·8-s + 2.72·9-s − 0.494·10-s − 0.864i·11-s − 0.965·12-s + 0.267·14-s − 1.35i·15-s + 0.250·16-s + 0.540·17-s − 1.92i·18-s + ⋯

Functional equation

Λ(s)=(2366s/2 ΓC(s) L(s)=((0.277+0.960i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2366s/2 ΓC(s+1/2) L(s)=((0.277+0.960i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 23662366    =    2⋅7⋅1322 \cdot 7 \cdot 13^{2}
Sign: 0.277+0.960i0.277 + 0.960i
Analytic conductor: 18.892618.8926
Root analytic conductor: 4.346564.34656
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2366(337,⋅)\chi_{2366} (337, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2366, ( :1/2), 0.277+0.960i)(2,\ 2366,\ (\ :1/2),\ 0.277 + 0.960i)

Particular Values

L(1)L(1) ≈\approx 3.7327398693.732739869
L(12)L(\frac12) ≈\approx 3.7327398693.732739869
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+iT 1 + iT
7 1−iT 1 - iT
13 1 1
good3 1−3.34T+3T2 1 - 3.34T + 3T^{2}
5 1+1.56iT−5T2 1 + 1.56iT - 5T^{2}
11 1+2.86iT−11T2 1 + 2.86iT - 11T^{2}
17 1−2.22T+17T2 1 - 2.22T + 17T^{2}
19 1−7.23iT−19T2 1 - 7.23iT - 19T^{2}
23 1−1.66T+23T2 1 - 1.66T + 23T^{2}
29 1−4.82T+29T2 1 - 4.82T + 29T^{2}
31 1−0.597iT−31T2 1 - 0.597iT - 31T^{2}
37 1−0.0385iT−37T2 1 - 0.0385iT - 37T^{2}
41 1+7.95iT−41T2 1 + 7.95iT - 41T^{2}
43 1+10.0T+43T2 1 + 10.0T + 43T^{2}
47 1+7.02iT−47T2 1 + 7.02iT - 47T^{2}
53 1+5.98T+53T2 1 + 5.98T + 53T^{2}
59 1+0.896iT−59T2 1 + 0.896iT - 59T^{2}
61 1+14.2T+61T2 1 + 14.2T + 61T^{2}
67 1+1.64iT−67T2 1 + 1.64iT - 67T^{2}
71 1−2.29iT−71T2 1 - 2.29iT - 71T^{2}
73 1+11.2iT−73T2 1 + 11.2iT - 73T^{2}
79 1−4.26T+79T2 1 - 4.26T + 79T^{2}
83 1−4.94iT−83T2 1 - 4.94iT - 83T^{2}
89 1−2.42iT−89T2 1 - 2.42iT - 89T^{2}
97 1−4.88iT−97T2 1 - 4.88iT - 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

−8.832500830060239259887084064618, −8.242482502169434364775952711522, −7.84671184623907950613199083689, −6.64460699015099268542264481461, −5.42318389656732978438018787941, −4.50587182708865069899432896339, −3.52680157627976291316565948654, −3.12922542856450717901637804147, −2.01000148615358992653645545767, −1.19613903470974563896250555108, 1.42283247614129531685830818695, 2.75767835067011089940151154748, 3.16666907709086992282922284898, 4.35968406373433591034895900829, 4.84571870813974388787381224248, 6.56575324605820833005178966439, 6.97208893212934821742637716390, 7.63613975529296236284647071677, 8.246752765615955366028492571493, 9.045173383973337820254348448691

Graph of the ZZ-function along the critical line