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Properties

Label 2-10e3-1.1-c1-0-4
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  − 3.01·3-s + 4.48·7-s + 6.10·9-s + 3.39·11-s − 6.49·13-s − 3.15·17-s − 1.11·19-s − 13.5·21-s + 4.98·23-s − 9.35·27-s + 0.354·29-s + 5.42·31-s − 10.2·33-s + 2.87·37-s + 19.6·39-s − 0.211·41-s − 0.847·43-s − 2.09·47-s + 13.0·49-s + 9.50·51-s + 9.06·53-s + 3.35·57-s + 3.46·59-s + 3.78·61-s + 27.3·63-s + 7.73·67-s − 15.0·69-s + ⋯
L(s)  = 1  − 1.74·3-s + 1.69·7-s + 2.03·9-s + 1.02·11-s − 1.80·13-s − 0.764·17-s − 0.254·19-s − 2.95·21-s + 1.04·23-s − 1.80·27-s + 0.0659·29-s + 0.974·31-s − 1.78·33-s + 0.471·37-s + 3.13·39-s − 0.0330·41-s − 0.129·43-s − 0.304·47-s + 1.87·49-s + 1.33·51-s + 1.24·53-s + 0.443·57-s + 0.451·59-s + 0.484·61-s + 3.44·63-s + 0.944·67-s − 1.81·69-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 1.0374987101.037498710
L(12)L(\frac12) ≈\approx 1.0374987101.037498710
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+3.01T+3T2 1 + 3.01T + 3T^{2}
7 1−4.48T+7T2 1 - 4.48T + 7T^{2}
11 1−3.39T+11T2 1 - 3.39T + 11T^{2}
13 1+6.49T+13T2 1 + 6.49T + 13T^{2}
17 1+3.15T+17T2 1 + 3.15T + 17T^{2}
19 1+1.11T+19T2 1 + 1.11T + 19T^{2}
23 1−4.98T+23T2 1 - 4.98T + 23T^{2}
29 1−0.354T+29T2 1 - 0.354T + 29T^{2}
31 1−5.42T+31T2 1 - 5.42T + 31T^{2}
37 1−2.87T+37T2 1 - 2.87T + 37T^{2}
41 1+0.211T+41T2 1 + 0.211T + 41T^{2}
43 1+0.847T+43T2 1 + 0.847T + 43T^{2}
47 1+2.09T+47T2 1 + 2.09T + 47T^{2}
53 1−9.06T+53T2 1 - 9.06T + 53T^{2}
59 1−3.46T+59T2 1 - 3.46T + 59T^{2}
61 1−3.78T+61T2 1 - 3.78T + 61T^{2}
67 1−7.73T+67T2 1 - 7.73T + 67T^{2}
71 1−0.0503T+71T2 1 - 0.0503T + 71T^{2}
73 1−7.62T+73T2 1 - 7.62T + 73T^{2}
79 1+8.60T+79T2 1 + 8.60T + 79T^{2}
83 1−14.7T+83T2 1 - 14.7T + 83T^{2}
89 1−9.84T+89T2 1 - 9.84T + 89T^{2}
97 1+10.7T+97T2 1 + 10.7T + 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

−10.22967107112080789203615807022, −9.307109874779712146883368797713, −8.194045146264873147356519406427, −7.16769043865944030227925991733, −6.63483316616915228765245408933, −5.46021367744418967326168074788, −4.81687255541543521495691303559, −4.31096963388915290782774110453, −2.15096214265904922896510184099, −0.900366202041063146447159861662, 0.900366202041063146447159861662, 2.15096214265904922896510184099, 4.31096963388915290782774110453, 4.81687255541543521495691303559, 5.46021367744418967326168074788, 6.63483316616915228765245408933, 7.16769043865944030227925991733, 8.194045146264873147356519406427, 9.307109874779712146883368797713, 10.22967107112080789203615807022

Graph of the ZZ-function along the critical line