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Properties

Label 2-4600-5.4-c1-0-25
Degree 22
Conductor 46004600
Sign 0.447−0.894i0.447 - 0.894i
Analytic cond. 36.731136.7311
Root an. cond. 6.060626.06062
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·7-s + 3·9-s − 6·11-s + 2i·13-s − 3i·17-s + 6·19-s − i·23-s − 3·29-s − 3·31-s + i·37-s + 9·41-s + 8i·43-s + 4i·47-s + 6·49-s − i·53-s + ⋯
L(s)  = 1  + 0.377i·7-s + 9-s − 1.80·11-s + 0.554i·13-s − 0.727i·17-s + 1.37·19-s − 0.208i·23-s − 0.557·29-s − 0.538·31-s + 0.164i·37-s + 1.40·41-s + 1.21i·43-s + 0.583i·47-s + 0.857·49-s − 0.137i·53-s + ⋯

Functional equation

Λ(s)=(4600s/2 ΓC(s) L(s)=((0.447−0.894i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(4600s/2 ΓC(s+1/2) L(s)=((0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 46004600    =    23⋅52⋅232^{3} \cdot 5^{2} \cdot 23
Sign: 0.447−0.894i0.447 - 0.894i
Analytic conductor: 36.731136.7311
Root analytic conductor: 6.060626.06062
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ4600(4049,⋅)\chi_{4600} (4049, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 4600, ( :1/2), 0.447−0.894i)(2,\ 4600,\ (\ :1/2),\ 0.447 - 0.894i)

Particular Values

L(1)L(1) ≈\approx 1.6332427031.633242703
L(12)L(\frac12) ≈\approx 1.6332427031.633242703
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
5 1 1
23 1+iT 1 + iT
good3 1−3T2 1 - 3T^{2}
7 1−iT−7T2 1 - iT - 7T^{2}
11 1+6T+11T2 1 + 6T + 11T^{2}
13 1−2iT−13T2 1 - 2iT - 13T^{2}
17 1+3iT−17T2 1 + 3iT - 17T^{2}
19 1−6T+19T2 1 - 6T + 19T^{2}
29 1+3T+29T2 1 + 3T + 29T^{2}
31 1+3T+31T2 1 + 3T + 31T^{2}
37 1−iT−37T2 1 - iT - 37T^{2}
41 1−9T+41T2 1 - 9T + 41T^{2}
43 1−8iT−43T2 1 - 8iT - 43T^{2}
47 1−4iT−47T2 1 - 4iT - 47T^{2}
53 1+iT−53T2 1 + iT - 53T^{2}
59 1+T+59T2 1 + T + 59T^{2}
61 1−8T+61T2 1 - 8T + 61T^{2}
67 1+7iT−67T2 1 + 7iT - 67T^{2}
71 1+5T+71T2 1 + 5T + 71T^{2}
73 1−6iT−73T2 1 - 6iT - 73T^{2}
79 1+79T2 1 + 79T^{2}
83 1−11iT−83T2 1 - 11iT - 83T^{2}
89 1+4T+89T2 1 + 4T + 89T^{2}
97 1−6iT−97T2 1 - 6iT - 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

−8.332500620763886995163268329317, −7.50760568985551690456235254807, −7.32455680407539074097898174167, −6.23306749822150298562826978274, −5.35202882263727020787012978441, −4.90217461302165383951306066785, −3.96414004902763441188966460948, −2.91103777254490309407546115060, −2.23419283449759969299113854842, −0.992050436900914754116202054484, 0.52597335362696331861488673489, 1.73684536868741167577433724392, 2.75557830537893932966884538548, 3.63685219384694221354548819077, 4.43718168230733038584775330490, 5.42755694932790245043144064116, 5.70561057142759781889338646595, 7.05327383951325524737312396510, 7.46721978945700448306441662723, 7.974684030038371362661499655364

Graph of the ZZ-function along the critical line