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Properties

Label 2-800-5.4-c3-0-37
Degree 22
Conductor 800800
Sign 0.894+0.447i0.894 + 0.447i
Analytic cond. 47.201547.2015
Root an. cond. 6.870336.87033
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  + 2i·3-s + 6i·7-s + 23·9-s + 60·11-s − 50i·13-s − 30i·17-s − 40·19-s − 12·21-s − 178i·23-s + 100i·27-s − 166·29-s + 20·31-s + 120i·33-s + 10i·37-s + 100·39-s + ⋯
L(s)  = 1  + 0.384i·3-s + 0.323i·7-s + 0.851·9-s + 1.64·11-s − 1.06i·13-s − 0.428i·17-s − 0.482·19-s − 0.124·21-s − 1.61i·23-s + 0.712i·27-s − 1.06·29-s + 0.115·31-s + 0.633i·33-s + 0.0444i·37-s + 0.410·39-s + ⋯

Functional equation

Λ(s)=(800s/2 ΓC(s) L(s)=((0.894+0.447i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(800s/2 ΓC(s+3/2) L(s)=((0.894+0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{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: 800800    =    25⋅522^{5} \cdot 5^{2}
Sign: 0.894+0.447i0.894 + 0.447i
Analytic conductor: 47.201547.2015
Root analytic conductor: 6.870336.87033
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ800(449,⋅)\chi_{800} (449, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 800, ( :3/2), 0.894+0.447i)(2,\ 800,\ (\ :3/2),\ 0.894 + 0.447i)

Particular Values

L(2)L(2) ≈\approx 2.3158277932.315827793
L(12)L(\frac12) ≈\approx 2.3158277932.315827793
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 1−2iT−27T2 1 - 2iT - 27T^{2}
7 1−6iT−343T2 1 - 6iT - 343T^{2}
11 1−60T+1.33e3T2 1 - 60T + 1.33e3T^{2}
13 1+50iT−2.19e3T2 1 + 50iT - 2.19e3T^{2}
17 1+30iT−4.91e3T2 1 + 30iT - 4.91e3T^{2}
19 1+40T+6.85e3T2 1 + 40T + 6.85e3T^{2}
23 1+178iT−1.21e4T2 1 + 178iT - 1.21e4T^{2}
29 1+166T+2.43e4T2 1 + 166T + 2.43e4T^{2}
31 1−20T+2.97e4T2 1 - 20T + 2.97e4T^{2}
37 1−10iT−5.06e4T2 1 - 10iT - 5.06e4T^{2}
41 1+250T+6.89e4T2 1 + 250T + 6.89e4T^{2}
43 1+142iT−7.95e4T2 1 + 142iT - 7.95e4T^{2}
47 1−214iT−1.03e5T2 1 - 214iT - 1.03e5T^{2}
53 1+490iT−1.48e5T2 1 + 490iT - 1.48e5T^{2}
59 1−800T+2.05e5T2 1 - 800T + 2.05e5T^{2}
61 1−250T+2.26e5T2 1 - 250T + 2.26e5T^{2}
67 1+774iT−3.00e5T2 1 + 774iT - 3.00e5T^{2}
71 1−100T+3.57e5T2 1 - 100T + 3.57e5T^{2}
73 1−230iT−3.89e5T2 1 - 230iT - 3.89e5T^{2}
79 1−1.32e3T+4.93e5T2 1 - 1.32e3T + 4.93e5T^{2}
83 1+982iT−5.71e5T2 1 + 982iT - 5.71e5T^{2}
89 1+874T+7.04e5T2 1 + 874T + 7.04e5T^{2}
97 1+310iT−9.12e5T2 1 + 310iT - 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

−9.816259057052212511119592589013, −9.032732472602482014352527687217, −8.263720276516943120000263714428, −7.08377117381034597765050253729, −6.40992542856802199887632623207, −5.28343637804395783537976304326, −4.29122235409380352279644887160, −3.47523689191275757292368106728, −2.05568073741559131502295398844, −0.70975310124047142595551272372, 1.18402841078320556941008993391, 1.92230138412185716536512001364, 3.78337878194168354118339153295, 4.19866277594699220204188359687, 5.63468931620937173438701790792, 6.78135814630263318874160444823, 7.04564683381188809825546020486, 8.213910924648798787863582097056, 9.254236435359113794638843290884, 9.722133654477880552358941816178

Graph of the ZZ-function along the critical line