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Properties

Label 2-600-1.1-c3-0-13
Degree 22
Conductor 600600
Sign 11
Analytic cond. 35.401135.4011
Root an. cond. 5.949885.94988
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 16·7-s + 9·9-s − 28·11-s + 26·13-s + 62·17-s − 68·19-s + 48·21-s + 208·23-s + 27·27-s − 58·29-s + 160·31-s − 84·33-s − 270·37-s + 78·39-s + 282·41-s − 76·43-s + 280·47-s − 87·49-s + 186·51-s + 210·53-s − 204·57-s + 196·59-s + 742·61-s + 144·63-s − 836·67-s + 624·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.863·7-s + 1/3·9-s − 0.767·11-s + 0.554·13-s + 0.884·17-s − 0.821·19-s + 0.498·21-s + 1.88·23-s + 0.192·27-s − 0.371·29-s + 0.926·31-s − 0.443·33-s − 1.19·37-s + 0.320·39-s + 1.07·41-s − 0.269·43-s + 0.868·47-s − 0.253·49-s + 0.510·51-s + 0.544·53-s − 0.474·57-s + 0.432·59-s + 1.55·61-s + 0.287·63-s − 1.52·67-s + 1.08·69-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 600600    =    23⋅3⋅522^{3} \cdot 3 \cdot 5^{2}
Sign: 11
Analytic conductor: 35.401135.4011
Root analytic conductor: 5.949885.94988
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 600, ( :3/2), 1)(2,\ 600,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 2.8435498772.843549877
L(12)L(\frac12) ≈\approx 2.8435498772.843549877
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−pT 1 - p T
5 1 1
good7 1−16T+p3T2 1 - 16 T + p^{3} T^{2}
11 1+28T+p3T2 1 + 28 T + p^{3} T^{2}
13 1−2pT+p3T2 1 - 2 p T + p^{3} T^{2}
17 1−62T+p3T2 1 - 62 T + p^{3} T^{2}
19 1+68T+p3T2 1 + 68 T + p^{3} T^{2}
23 1−208T+p3T2 1 - 208 T + p^{3} T^{2}
29 1+2pT+p3T2 1 + 2 p T + p^{3} T^{2}
31 1−160T+p3T2 1 - 160 T + p^{3} T^{2}
37 1+270T+p3T2 1 + 270 T + p^{3} T^{2}
41 1−282T+p3T2 1 - 282 T + p^{3} T^{2}
43 1+76T+p3T2 1 + 76 T + p^{3} T^{2}
47 1−280T+p3T2 1 - 280 T + p^{3} T^{2}
53 1−210T+p3T2 1 - 210 T + p^{3} T^{2}
59 1−196T+p3T2 1 - 196 T + p^{3} T^{2}
61 1−742T+p3T2 1 - 742 T + p^{3} T^{2}
67 1+836T+p3T2 1 + 836 T + p^{3} T^{2}
71 1+504T+p3T2 1 + 504 T + p^{3} T^{2}
73 1−1062T+p3T2 1 - 1062 T + p^{3} T^{2}
79 1−768T+p3T2 1 - 768 T + p^{3} T^{2}
83 1−1052T+p3T2 1 - 1052 T + p^{3} T^{2}
89 1+726T+p3T2 1 + 726 T + p^{3} T^{2}
97 1−1406T+p3T2 1 - 1406 T + p^{3} T^{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

−10.40315258013460502497575714427, −9.231469528224753965378959459148, −8.449190075546532196723925377173, −7.77635714893283745088183976587, −6.83701489903676173234531684575, −5.53609398247293713209504942330, −4.67871952250030086479151753184, −3.47048686458413917204240061837, −2.34248148855115171478782764637, −1.04256043740078889423438437899, 1.04256043740078889423438437899, 2.34248148855115171478782764637, 3.47048686458413917204240061837, 4.67871952250030086479151753184, 5.53609398247293713209504942330, 6.83701489903676173234531684575, 7.77635714893283745088183976587, 8.449190075546532196723925377173, 9.231469528224753965378959459148, 10.40315258013460502497575714427

Graph of the ZZ-function along the critical line