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Properties

Label 2-600-5.4-c3-0-10
Degree 22
Conductor 600600
Sign 0.447−0.894i0.447 - 0.894i
Analytic cond. 35.401135.4011
Root an. cond. 5.949885.94988
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3i·3-s + 4i·7-s − 9·9-s + 72·11-s + 6i·13-s + 38i·17-s − 52·19-s − 12·21-s − 152i·23-s − 27i·27-s + 78·29-s + 120·31-s + 216i·33-s − 150i·37-s − 18·39-s + ⋯
L(s)  = 1  + 0.577i·3-s + 0.215i·7-s − 0.333·9-s + 1.97·11-s + 0.128i·13-s + 0.542i·17-s − 0.627·19-s − 0.124·21-s − 1.37i·23-s − 0.192i·27-s + 0.499·29-s + 0.695·31-s + 1.13i·33-s − 0.666i·37-s − 0.0739·39-s + ⋯

Functional equation

Λ(s)=(600s/2 ΓC(s) L(s)=((0.447−0.894i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(600s/2 ΓC(s+3/2) L(s)=((0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 600600    =    23⋅3⋅522^{3} \cdot 3 \cdot 5^{2}
Sign: 0.447−0.894i0.447 - 0.894i
Analytic conductor: 35.401135.4011
Root analytic conductor: 5.949885.94988
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ600(49,⋅)\chi_{600} (49, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 600, ( :3/2), 0.447−0.894i)(2,\ 600,\ (\ :3/2),\ 0.447 - 0.894i)

Particular Values

L(2)L(2) ≈\approx 2.1480523482.148052348
L(12)L(\frac12) ≈\approx 2.1480523482.148052348
L(52)L(\frac{5}{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 1−3iT 1 - 3iT
5 1 1
good7 1−4iT−343T2 1 - 4iT - 343T^{2}
11 1−72T+1.33e3T2 1 - 72T + 1.33e3T^{2}
13 1−6iT−2.19e3T2 1 - 6iT - 2.19e3T^{2}
17 1−38iT−4.91e3T2 1 - 38iT - 4.91e3T^{2}
19 1+52T+6.85e3T2 1 + 52T + 6.85e3T^{2}
23 1+152iT−1.21e4T2 1 + 152iT - 1.21e4T^{2}
29 1−78T+2.43e4T2 1 - 78T + 2.43e4T^{2}
31 1−120T+2.97e4T2 1 - 120T + 2.97e4T^{2}
37 1+150iT−5.06e4T2 1 + 150iT - 5.06e4T^{2}
41 1−362T+6.89e4T2 1 - 362T + 6.89e4T^{2}
43 1−484iT−7.95e4T2 1 - 484iT - 7.95e4T^{2}
47 1−280iT−1.03e5T2 1 - 280iT - 1.03e5T^{2}
53 1−670iT−1.48e5T2 1 - 670iT - 1.48e5T^{2}
59 1+696T+2.05e5T2 1 + 696T + 2.05e5T^{2}
61 1−222T+2.26e5T2 1 - 222T + 2.26e5T^{2}
67 1+4iT−3.00e5T2 1 + 4iT - 3.00e5T^{2}
71 1−96T+3.57e5T2 1 - 96T + 3.57e5T^{2}
73 1+178iT−3.89e5T2 1 + 178iT - 3.89e5T^{2}
79 1−632T+4.93e5T2 1 - 632T + 4.93e5T^{2}
83 1−612iT−5.71e5T2 1 - 612iT - 5.71e5T^{2}
89 1+994T+7.04e5T2 1 + 994T + 7.04e5T^{2}
97 1−1.63e3iT−9.12e5T2 1 - 1.63e3iT - 9.12e5T^{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

−10.47177426064802263608151346761, −9.347945959485985767297050085268, −8.938881734441480588555998880665, −7.913352370001075159685331044057, −6.52540282771831746062670303778, −6.08985261807926313753308484821, −4.53397163324778011718774849468, −4.00563703197745381294656309836, −2.62445535050150252813210834323, −1.11661554429713011183707908193, 0.77255853202351811059780156652, 1.88234489221942233959947996107, 3.38064279740927582896832935625, 4.36417776808573105888307903932, 5.69668832742928822351224076743, 6.64125801374809805071914994705, 7.25289629556677057149254609618, 8.402252492826185807553042885624, 9.171320395518647947085934709538, 10.00740828160630579661965223252

Graph of the ZZ-function along the critical line