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Properties

Label 2-4002-1.1-c1-0-13
Degree 22
Conductor 40024002
Sign 11
Analytic cond. 31.956131.9561
Root an. cond. 5.652975.65297
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-s + 3-s + 4-s − 1.56·5-s − 6-s − 1.09·7-s − 8-s + 9-s + 1.56·10-s + 4.17·11-s + 12-s − 2.33·13-s + 1.09·14-s − 1.56·15-s + 16-s − 0.0496·17-s − 18-s + 2.48·19-s − 1.56·20-s − 1.09·21-s − 4.17·22-s + 23-s − 24-s − 2.56·25-s + 2.33·26-s + 27-s − 1.09·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.698·5-s − 0.408·6-s − 0.412·7-s − 0.353·8-s + 0.333·9-s + 0.493·10-s + 1.25·11-s + 0.288·12-s − 0.648·13-s + 0.291·14-s − 0.403·15-s + 0.250·16-s − 0.0120·17-s − 0.235·18-s + 0.569·19-s − 0.349·20-s − 0.238·21-s − 0.890·22-s + 0.208·23-s − 0.204·24-s − 0.512·25-s + 0.458·26-s + 0.192·27-s − 0.206·28-s + ⋯

Functional equation

Λ(s)=(4002s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4002s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 40024002    =    2⋅3⋅23⋅292 \cdot 3 \cdot 23 \cdot 29
Sign: 11
Analytic conductor: 31.956131.9561
Root analytic conductor: 5.652975.65297
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4002, ( :1/2), 1)(2,\ 4002,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.3833764591.383376459
L(12)L(\frac12) ≈\approx 1.3833764591.383376459
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+T 1 + T
3 1−T 1 - T
23 1−T 1 - T
29 1−T 1 - T
good5 1+1.56T+5T2 1 + 1.56T + 5T^{2}
7 1+1.09T+7T2 1 + 1.09T + 7T^{2}
11 1−4.17T+11T2 1 - 4.17T + 11T^{2}
13 1+2.33T+13T2 1 + 2.33T + 13T^{2}
17 1+0.0496T+17T2 1 + 0.0496T + 17T^{2}
19 1−2.48T+19T2 1 - 2.48T + 19T^{2}
31 1+0.863T+31T2 1 + 0.863T + 31T^{2}
37 1−9.50T+37T2 1 - 9.50T + 37T^{2}
41 1+10.0T+41T2 1 + 10.0T + 41T^{2}
43 1−1.25T+43T2 1 - 1.25T + 43T^{2}
47 1−0.476T+47T2 1 - 0.476T + 47T^{2}
53 1−12.0T+53T2 1 - 12.0T + 53T^{2}
59 1−1.02T+59T2 1 - 1.02T + 59T^{2}
61 1−12.5T+61T2 1 - 12.5T + 61T^{2}
67 1+5.94T+67T2 1 + 5.94T + 67T^{2}
71 1−4.49T+71T2 1 - 4.49T + 71T^{2}
73 1+0.0841T+73T2 1 + 0.0841T + 73T^{2}
79 1+12.4T+79T2 1 + 12.4T + 79T^{2}
83 1−2.88T+83T2 1 - 2.88T + 83T^{2}
89 1−6.00T+89T2 1 - 6.00T + 89T^{2}
97 1+1.94T+97T2 1 + 1.94T + 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

−8.483012883296788311330797106196, −7.79465497337517114363767956676, −7.11342669809506716150550606157, −6.59064063845597889790148194142, −5.59573972784434273316741651444, −4.44355445867998118212552951991, −3.70981552321667188876355036032, −2.95001307629233913367159236299, −1.89331894051472742213117147580, −0.73263615379500269157525798365, 0.73263615379500269157525798365, 1.89331894051472742213117147580, 2.95001307629233913367159236299, 3.70981552321667188876355036032, 4.44355445867998118212552951991, 5.59573972784434273316741651444, 6.59064063845597889790148194142, 7.11342669809506716150550606157, 7.79465497337517114363767956676, 8.483012883296788311330797106196

Graph of the ZZ-function along the critical line