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Properties

Label 2-10e3-1.1-c1-0-1
Degree 22
Conductor 10001000
Sign 11
Analytic cond. 7.985047.98504
Root an. cond. 2.825782.82578
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  − 2.39·3-s − 1.13·7-s + 2.75·9-s − 2.01·11-s − 2.26·13-s + 1.08·17-s + 5.58·19-s + 2.72·21-s − 8.33·23-s + 0.591·27-s + 9.11·29-s − 8.75·31-s + 4.83·33-s + 2.54·37-s + 5.42·39-s + 9.82·41-s + 2.91·43-s + 2.09·47-s − 5.71·49-s − 2.59·51-s + 10.5·53-s − 13.3·57-s + 5.53·59-s − 1.63·61-s − 3.12·63-s + 10.5·67-s + 19.9·69-s + ⋯
L(s)  = 1  − 1.38·3-s − 0.429·7-s + 0.917·9-s − 0.608·11-s − 0.627·13-s + 0.262·17-s + 1.28·19-s + 0.594·21-s − 1.73·23-s + 0.113·27-s + 1.69·29-s − 1.57·31-s + 0.842·33-s + 0.418·37-s + 0.869·39-s + 1.53·41-s + 0.444·43-s + 0.304·47-s − 0.815·49-s − 0.364·51-s + 1.44·53-s − 1.77·57-s + 0.720·59-s − 0.209·61-s − 0.393·63-s + 1.29·67-s + 2.40·69-s + ⋯

Functional equation

Λ(s)=(1000s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1000s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10001000    =    23⋅532^{3} \cdot 5^{3}
Sign: 11
Analytic conductor: 7.985047.98504
Root analytic conductor: 2.825782.82578
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1000, ( :1/2), 1)(2,\ 1000,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.72871799210.7287179921
L(12)L(\frac12) ≈\approx 0.72871799210.7287179921
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+2.39T+3T2 1 + 2.39T + 3T^{2}
7 1+1.13T+7T2 1 + 1.13T + 7T^{2}
11 1+2.01T+11T2 1 + 2.01T + 11T^{2}
13 1+2.26T+13T2 1 + 2.26T + 13T^{2}
17 1−1.08T+17T2 1 - 1.08T + 17T^{2}
19 1−5.58T+19T2 1 - 5.58T + 19T^{2}
23 1+8.33T+23T2 1 + 8.33T + 23T^{2}
29 1−9.11T+29T2 1 - 9.11T + 29T^{2}
31 1+8.75T+31T2 1 + 8.75T + 31T^{2}
37 1−2.54T+37T2 1 - 2.54T + 37T^{2}
41 1−9.82T+41T2 1 - 9.82T + 41T^{2}
43 1−2.91T+43T2 1 - 2.91T + 43T^{2}
47 1−2.09T+47T2 1 - 2.09T + 47T^{2}
53 1−10.5T+53T2 1 - 10.5T + 53T^{2}
59 1−5.53T+59T2 1 - 5.53T + 59T^{2}
61 1+1.63T+61T2 1 + 1.63T + 61T^{2}
67 1−10.5T+67T2 1 - 10.5T + 67T^{2}
71 1−12.9T+71T2 1 - 12.9T + 71T^{2}
73 1−13.2T+73T2 1 - 13.2T + 73T^{2}
79 1−6.84T+79T2 1 - 6.84T + 79T^{2}
83 1−2.81T+83T2 1 - 2.81T + 83T^{2}
89 1+15.1T+89T2 1 + 15.1T + 89T^{2}
97 1−18.2T+97T2 1 - 18.2T + 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

−10.05405447131922687042495649100, −9.485929206143727185686810689449, −8.133786011535845071559390579758, −7.33362621496411984232201860138, −6.41942504501568628932123233917, −5.61998745621760884168344747670, −5.02543349921269280279835990871, −3.85183112850723098709754305594, −2.47559031039543173516635473788, −0.69440023213056820669337109707, 0.69440023213056820669337109707, 2.47559031039543173516635473788, 3.85183112850723098709754305594, 5.02543349921269280279835990871, 5.61998745621760884168344747670, 6.41942504501568628932123233917, 7.33362621496411984232201860138, 8.133786011535845071559390579758, 9.485929206143727185686810689449, 10.05405447131922687042495649100

Graph of the ZZ-function along the critical line