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Properties

Label 2-600-40.29-c1-0-0
Degree 22
Conductor 600600
Sign −0.929−0.368i-0.929 - 0.368i
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  + (−1.24 − 0.671i)2-s − 3-s + (1.09 + 1.67i)4-s + (1.24 + 0.671i)6-s + 4.68i·7-s + (−0.244 − 2.81i)8-s + 9-s + 2.29i·11-s + (−1.09 − 1.67i)12-s − 4.97·13-s + (3.14 − 5.83i)14-s + (−1.58 + 3.67i)16-s − 2.97i·17-s + (−1.24 − 0.671i)18-s − 2.68i·19-s + ⋯
L(s)  = 1  + (−0.880 − 0.474i)2-s − 0.577·3-s + (0.549 + 0.835i)4-s + (0.508 + 0.274i)6-s + 1.77i·7-s + (−0.0864 − 0.996i)8-s + 0.333·9-s + 0.691i·11-s + (−0.317 − 0.482i)12-s − 1.38·13-s + (0.840 − 1.55i)14-s + (−0.396 + 0.917i)16-s − 0.722i·17-s + (−0.293 − 0.158i)18-s − 0.616i·19-s + ⋯

Functional equation

Λ(s)=(600s/2 ΓC(s) L(s)=((−0.929−0.368i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(600s/2 ΓC(s+1/2) L(s)=((−0.929−0.368i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 600600    =    23⋅3⋅522^{3} \cdot 3 \cdot 5^{2}
Sign: −0.929−0.368i-0.929 - 0.368i
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), −0.929−0.368i)(2,\ 600,\ (\ :1/2),\ -0.929 - 0.368i)

Particular Values

L(1)L(1) ≈\approx 0.0386860+0.202769i0.0386860 + 0.202769i
L(12)L(\frac12) ≈\approx 0.0386860+0.202769i0.0386860 + 0.202769i
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+(1.24+0.671i)T 1 + (1.24 + 0.671i)T
3 1+T 1 + T
5 1 1
good7 1−4.68iT−7T2 1 - 4.68iT - 7T^{2}
11 1−2.29iT−11T2 1 - 2.29iT - 11T^{2}
13 1+4.97T+13T2 1 + 4.97T + 13T^{2}
17 1+2.97iT−17T2 1 + 2.97iT - 17T^{2}
19 1+2.68iT−19T2 1 + 2.68iT - 19T^{2}
23 1+2.68iT−23T2 1 + 2.68iT - 23T^{2}
29 1+2iT−29T2 1 + 2iT - 29T^{2}
31 1+6.97T+31T2 1 + 6.97T + 31T^{2}
37 1+4.39T+37T2 1 + 4.39T + 37T^{2}
41 1+11.3T+41T2 1 + 11.3T + 41T^{2}
43 1−9.37T+43T2 1 - 9.37T + 43T^{2}
47 1−7.27iT−47T2 1 - 7.27iT - 47T^{2}
53 1+2T+53T2 1 + 2T + 53T^{2}
59 1+1.70iT−59T2 1 + 1.70iT - 59T^{2}
61 1−4.58iT−61T2 1 - 4.58iT - 61T^{2}
67 1+4T+67T2 1 + 4T + 67T^{2}
71 1−0.585T+71T2 1 - 0.585T + 71T^{2}
73 1−6iT−73T2 1 - 6iT - 73T^{2}
79 1+1.02T+79T2 1 + 1.02T + 79T^{2}
83 1+13.3T+83T2 1 + 13.3T + 83T^{2}
89 1+3.37T+89T2 1 + 3.37T + 89T^{2}
97 1+3.95iT−97T2 1 + 3.95iT - 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

−11.11157654784755477245954175295, −10.02026040176976769975526345745, −9.380329199787679926697331365580, −8.706128994632291139668886243773, −7.53337282257966056634324225003, −6.77877885429962459328927523297, −5.57605520652319942218035698053, −4.63920355519819174369012170356, −2.84517853161744640264327861742, −2.02945077535826425175472285014, 0.15903604181634495935897673885, 1.59766349238880785080994304301, 3.61138327812209258684653296657, 4.89450136833995656433304553409, 5.88253842929194725291443068078, 7.02075718292190778703267685126, 7.40233120213116686245365138930, 8.393972131618795726704724514302, 9.600100498880543140923733283646, 10.33167703261758230817142898171

Graph of the ZZ-function along the critical line