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Properties

Label 2-800-1.1-c1-0-8
Degree 22
Conductor 800800
Sign 11
Analytic cond. 6.388036.38803
Root an. cond. 2.527452.52745
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.82·3-s − 2.82·7-s + 5.00·9-s + 5.65·11-s + 2·13-s − 2·17-s − 8.00·21-s + 2.82·23-s + 5.65·27-s + 6·29-s − 5.65·31-s + 16.0·33-s + 10·37-s + 5.65·39-s + 2·41-s − 8.48·43-s − 2.82·47-s + 1.00·49-s − 5.65·51-s − 6·53-s − 11.3·59-s − 2·61-s − 14.1·63-s − 2.82·67-s + 8.00·69-s − 5.65·71-s + 6·73-s + ⋯
L(s)  = 1  + 1.63·3-s − 1.06·7-s + 1.66·9-s + 1.70·11-s + 0.554·13-s − 0.485·17-s − 1.74·21-s + 0.589·23-s + 1.08·27-s + 1.11·29-s − 1.01·31-s + 2.78·33-s + 1.64·37-s + 0.905·39-s + 0.312·41-s − 1.29·43-s − 0.412·47-s + 0.142·49-s − 0.792·51-s − 0.824·53-s − 1.47·59-s − 0.256·61-s − 1.78·63-s − 0.345·67-s + 0.963·69-s − 0.671·71-s + 0.702·73-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 800800    =    25⋅522^{5} \cdot 5^{2}
Sign: 11
Analytic conductor: 6.388036.38803
Root analytic conductor: 2.527452.52745
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 800, ( :1/2), 1)(2,\ 800,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.6472632992.647263299
L(12)L(\frac12) ≈\approx 2.6472632992.647263299
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
5 1 1
good3 1−2.82T+3T2 1 - 2.82T + 3T^{2}
7 1+2.82T+7T2 1 + 2.82T + 7T^{2}
11 1−5.65T+11T2 1 - 5.65T + 11T^{2}
13 1−2T+13T2 1 - 2T + 13T^{2}
17 1+2T+17T2 1 + 2T + 17T^{2}
19 1+19T2 1 + 19T^{2}
23 1−2.82T+23T2 1 - 2.82T + 23T^{2}
29 1−6T+29T2 1 - 6T + 29T^{2}
31 1+5.65T+31T2 1 + 5.65T + 31T^{2}
37 1−10T+37T2 1 - 10T + 37T^{2}
41 1−2T+41T2 1 - 2T + 41T^{2}
43 1+8.48T+43T2 1 + 8.48T + 43T^{2}
47 1+2.82T+47T2 1 + 2.82T + 47T^{2}
53 1+6T+53T2 1 + 6T + 53T^{2}
59 1+11.3T+59T2 1 + 11.3T + 59T^{2}
61 1+2T+61T2 1 + 2T + 61T^{2}
67 1+2.82T+67T2 1 + 2.82T + 67T^{2}
71 1+5.65T+71T2 1 + 5.65T + 71T^{2}
73 1−6T+73T2 1 - 6T + 73T^{2}
79 1+11.3T+79T2 1 + 11.3T + 79T^{2}
83 1−2.82T+83T2 1 - 2.82T + 83T^{2}
89 1−10T+89T2 1 - 10T + 89T^{2}
97 1+2T+97T2 1 + 2T + 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

−9.841080230976107745397994889488, −9.257165327628810739427304201856, −8.810139012028636033415322418102, −7.85576160174656386917305477794, −6.80132253934060211074539515319, −6.24697705176838528166337922019, −4.43821910497114992227960235246, −3.58906732979012245646385896249, −2.88298824871583873851569545601, −1.49471919423093806973675175961, 1.49471919423093806973675175961, 2.88298824871583873851569545601, 3.58906732979012245646385896249, 4.43821910497114992227960235246, 6.24697705176838528166337922019, 6.80132253934060211074539515319, 7.85576160174656386917305477794, 8.810139012028636033415322418102, 9.257165327628810739427304201856, 9.841080230976107745397994889488

Graph of the ZZ-function along the critical line