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Properties

Label 2-800-1.1-c3-0-7
Degree 22
Conductor 800800
Sign 11
Analytic cond. 47.201547.2015
Root an. cond. 6.870336.87033
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  − 2·3-s + 6·7-s − 23·9-s − 60·11-s − 50·13-s + 30·17-s − 40·19-s − 12·21-s + 178·23-s + 100·27-s + 166·29-s − 20·31-s + 120·33-s − 10·37-s + 100·39-s − 250·41-s + 142·43-s + 214·47-s − 307·49-s − 60·51-s − 490·53-s + 80·57-s + 800·59-s + 250·61-s − 138·63-s − 774·67-s − 356·69-s + ⋯
L(s)  = 1  − 0.384·3-s + 0.323·7-s − 0.851·9-s − 1.64·11-s − 1.06·13-s + 0.428·17-s − 0.482·19-s − 0.124·21-s + 1.61·23-s + 0.712·27-s + 1.06·29-s − 0.115·31-s + 0.633·33-s − 0.0444·37-s + 0.410·39-s − 0.952·41-s + 0.503·43-s + 0.664·47-s − 0.895·49-s − 0.164·51-s − 1.26·53-s + 0.185·57-s + 1.76·59-s + 0.524·61-s − 0.275·63-s − 1.41·67-s − 0.621·69-s + ⋯

Functional equation

Λ(s)=(800s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(800s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 800800    =    25⋅522^{5} \cdot 5^{2}
Sign: 11
Analytic conductor: 47.201547.2015
Root analytic conductor: 6.870336.87033
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 800, ( :3/2), 1)(2,\ 800,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.0916916911.091691691
L(12)L(\frac12) ≈\approx 1.0916916911.091691691
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
5 1 1
good3 1+2T+p3T2 1 + 2 T + p^{3} T^{2}
7 1−6T+p3T2 1 - 6 T + p^{3} T^{2}
11 1+60T+p3T2 1 + 60 T + p^{3} T^{2}
13 1+50T+p3T2 1 + 50 T + p^{3} T^{2}
17 1−30T+p3T2 1 - 30 T + p^{3} T^{2}
19 1+40T+p3T2 1 + 40 T + p^{3} T^{2}
23 1−178T+p3T2 1 - 178 T + p^{3} T^{2}
29 1−166T+p3T2 1 - 166 T + p^{3} T^{2}
31 1+20T+p3T2 1 + 20 T + p^{3} T^{2}
37 1+10T+p3T2 1 + 10 T + p^{3} T^{2}
41 1+250T+p3T2 1 + 250 T + p^{3} T^{2}
43 1−142T+p3T2 1 - 142 T + p^{3} T^{2}
47 1−214T+p3T2 1 - 214 T + p^{3} T^{2}
53 1+490T+p3T2 1 + 490 T + p^{3} T^{2}
59 1−800T+p3T2 1 - 800 T + p^{3} T^{2}
61 1−250T+p3T2 1 - 250 T + p^{3} T^{2}
67 1+774T+p3T2 1 + 774 T + p^{3} T^{2}
71 1+100T+p3T2 1 + 100 T + p^{3} T^{2}
73 1−230T+p3T2 1 - 230 T + p^{3} T^{2}
79 1−1320T+p3T2 1 - 1320 T + p^{3} T^{2}
83 1−982T+p3T2 1 - 982 T + p^{3} T^{2}
89 1−874T+p3T2 1 - 874 T + p^{3} T^{2}
97 1−310T+p3T2 1 - 310 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

−10.08421165210963073679623468747, −8.950415919407380678132044503976, −8.123621457200789690700512275096, −7.38136997373497635068579341322, −6.32378023986708973375165579652, −5.14530881383801815440747322818, −4.92749321527704198485452966509, −3.15142490379693847056066890327, −2.33789161253966981763205654397, −0.56866515580141363377599688471, 0.56866515580141363377599688471, 2.33789161253966981763205654397, 3.15142490379693847056066890327, 4.92749321527704198485452966509, 5.14530881383801815440747322818, 6.32378023986708973375165579652, 7.38136997373497635068579341322, 8.123621457200789690700512275096, 8.950415919407380678132044503976, 10.08421165210963073679623468747

Graph of the ZZ-function along the critical line