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Properties

Label 2-600-5.4-c3-0-8
Degree 22
Conductor 600600
Sign 0.894−0.447i0.894 - 0.447i
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 + 5i·7-s − 9·9-s + 14·11-s − i·13-s + 46i·17-s − 19·19-s + 15·21-s + 46i·23-s + 27i·27-s − 14·29-s + 133·31-s − 42i·33-s + 258i·37-s − 3·39-s + ⋯
L(s)  = 1  − 0.577i·3-s + 0.269i·7-s − 0.333·9-s + 0.383·11-s − 0.0213i·13-s + 0.656i·17-s − 0.229·19-s + 0.155·21-s + 0.417i·23-s + 0.192i·27-s − 0.0896·29-s + 0.770·31-s − 0.221i·33-s + 1.14i·37-s − 0.0123·39-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 600600    =    23⋅3⋅522^{3} \cdot 3 \cdot 5^{2}
Sign: 0.894−0.447i0.894 - 0.447i
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), 0.894−0.447i)(2,\ 600,\ (\ :3/2),\ 0.894 - 0.447i)

Particular Values

L(2)L(2) ≈\approx 1.7723165651.772316565
L(12)L(\frac12) ≈\approx 1.7723165651.772316565
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 1−5iT−343T2 1 - 5iT - 343T^{2}
11 1−14T+1.33e3T2 1 - 14T + 1.33e3T^{2}
13 1+iT−2.19e3T2 1 + iT - 2.19e3T^{2}
17 1−46iT−4.91e3T2 1 - 46iT - 4.91e3T^{2}
19 1+19T+6.85e3T2 1 + 19T + 6.85e3T^{2}
23 1−46iT−1.21e4T2 1 - 46iT - 1.21e4T^{2}
29 1+14T+2.43e4T2 1 + 14T + 2.43e4T^{2}
31 1−133T+2.97e4T2 1 - 133T + 2.97e4T^{2}
37 1−258iT−5.06e4T2 1 - 258iT - 5.06e4T^{2}
41 1−84T+6.89e4T2 1 - 84T + 6.89e4T^{2}
43 1−167iT−7.95e4T2 1 - 167iT - 7.95e4T^{2}
47 1−410iT−1.03e5T2 1 - 410iT - 1.03e5T^{2}
53 1+456iT−1.48e5T2 1 + 456iT - 1.48e5T^{2}
59 1−194T+2.05e5T2 1 - 194T + 2.05e5T^{2}
61 1+17T+2.26e5T2 1 + 17T + 2.26e5T^{2}
67 1−653iT−3.00e5T2 1 - 653iT - 3.00e5T^{2}
71 1−828T+3.57e5T2 1 - 828T + 3.57e5T^{2}
73 1+570iT−3.89e5T2 1 + 570iT - 3.89e5T^{2}
79 1−552T+4.93e5T2 1 - 552T + 4.93e5T^{2}
83 1+142iT−5.71e5T2 1 + 142iT - 5.71e5T^{2}
89 1−1.10e3T+7.04e5T2 1 - 1.10e3T + 7.04e5T^{2}
97 1−841iT−9.12e5T2 1 - 841iT - 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

−10.33632274258952882546110309110, −9.388110656755244807452141179795, −8.478318440346804837615389793335, −7.73148481151146225187159570789, −6.64919088844804273824691245604, −5.96241722043928980122202333366, −4.78930129485965841094195935731, −3.55071016051753500917026201096, −2.29904763295894959878539969108, −1.06905548052865161491382716048, 0.61610103676195062383139678494, 2.32168395748992946324983042550, 3.60669195043565804127643780183, 4.50809847561764285088880805692, 5.52111305824844172373223341601, 6.58902989762741280644121028035, 7.52872087305653300171032780579, 8.612421736483666771433280564708, 9.342480210181074280669813404427, 10.22257743998145646090133329610

Graph of the ZZ-function along the critical line