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Properties

Label 2-600-40.29-c1-0-18
Degree 22
Conductor 600600
Sign −0.717+0.696i-0.717 + 0.696i
Analytic cond. 4.791024.79102
Root an. cond. 2.188842.18884
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  + (−0.144 − 1.40i)2-s − 3-s + (−1.95 + 0.406i)4-s + (0.144 + 1.40i)6-s + 3.62i·7-s + (0.855 + 2.69i)8-s + 9-s − 6.20i·11-s + (1.95 − 0.406i)12-s − 0.578·13-s + (5.10 − 0.524i)14-s + (3.66 − 1.59i)16-s − 1.42i·17-s + (−0.144 − 1.40i)18-s − 5.62i·19-s + ⋯
L(s)  = 1  + (−0.102 − 0.994i)2-s − 0.577·3-s + (−0.979 + 0.203i)4-s + (0.0590 + 0.574i)6-s + 1.37i·7-s + (0.302 + 0.953i)8-s + 0.333·9-s − 1.87i·11-s + (0.565 − 0.117i)12-s − 0.160·13-s + (1.36 − 0.140i)14-s + (0.917 − 0.398i)16-s − 0.344i·17-s + (−0.0340 − 0.331i)18-s − 1.29i·19-s + ⋯

Functional equation

Λ(s)=(600s/2 ΓC(s) L(s)=((−0.717+0.696i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.717 + 0.696i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(600s/2 ΓC(s+1/2) L(s)=((−0.717+0.696i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.717 + 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 600600    =    23⋅3⋅522^{3} \cdot 3 \cdot 5^{2}
Sign: −0.717+0.696i-0.717 + 0.696i
Analytic conductor: 4.791024.79102
Root analytic conductor: 2.188842.18884
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ600(349,⋅)\chi_{600} (349, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 600, ( :1/2), −0.717+0.696i)(2,\ 600,\ (\ :1/2),\ -0.717 + 0.696i)

Particular Values

L(1)L(1) ≈\approx 0.300045−0.739497i0.300045 - 0.739497i
L(12)L(\frac12) ≈\approx 0.300045−0.739497i0.300045 - 0.739497i
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+(0.144+1.40i)T 1 + (0.144 + 1.40i)T
3 1+T 1 + T
5 1 1
good7 1−3.62iT−7T2 1 - 3.62iT - 7T^{2}
11 1+6.20iT−11T2 1 + 6.20iT - 11T^{2}
13 1+0.578T+13T2 1 + 0.578T + 13T^{2}
17 1+1.42iT−17T2 1 + 1.42iT - 17T^{2}
19 1+5.62iT−19T2 1 + 5.62iT - 19T^{2}
23 1+5.62iT−23T2 1 + 5.62iT - 23T^{2}
29 1−2iT−29T2 1 - 2iT - 29T^{2}
31 1+2.57T+31T2 1 + 2.57T + 31T^{2}
37 1−7.83T+37T2 1 - 7.83T + 37T^{2}
41 1−5.25T+41T2 1 - 5.25T + 41T^{2}
43 1+7.25T+43T2 1 + 7.25T + 43T^{2}
47 1+6.78iT−47T2 1 + 6.78iT - 47T^{2}
53 1+2T+53T2 1 + 2T + 53T^{2}
59 1+2.20iT−59T2 1 + 2.20iT - 59T^{2}
61 1+12.4iT−61T2 1 + 12.4iT - 61T^{2}
67 1+4T+67T2 1 + 4T + 67T^{2}
71 1−8.41T+71T2 1 - 8.41T + 71T^{2}
73 1+6iT−73T2 1 + 6iT - 73T^{2}
79 1+5.42T+79T2 1 + 5.42T + 79T^{2}
83 1−3.25T+83T2 1 - 3.25T + 83T^{2}
89 1−13.2T+89T2 1 - 13.2T + 89T^{2}
97 1+4.84iT−97T2 1 + 4.84iT - 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

−10.64675645213611373785847268377, −9.409860630105957912173617213996, −8.836255929669702365732464080210, −8.070886150020945909963537640349, −6.46401306264438788070095442518, −5.55728717117692810265656600414, −4.77537189873276606493055123896, −3.29705382881241156709347973515, −2.37782268559277801117909772703, −0.54388433427110706472065004062, 1.41435310844905832860546651901, 3.93153066843160318929114544702, 4.51384057681432099411117707924, 5.62127773430141346814980798220, 6.64609790711730922408455178439, 7.45922572358450266777781150226, 7.85476970944891725221182309500, 9.473566897626017550280921785378, 9.967124015259165221254840775192, 10.68478174294418379747816060220

Graph of the ZZ-function along the critical line