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Properties

Label 2-4002-1.1-c1-0-51
Degree 22
Conductor 40024002
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s − 0.694·5-s + 6-s + 1.42·7-s − 8-s + 9-s + 0.694·10-s − 0.355·11-s − 12-s − 3.56·13-s − 1.42·14-s + 0.694·15-s + 16-s + 1.04·17-s − 18-s + 5.27·19-s − 0.694·20-s − 1.42·21-s + 0.355·22-s + 23-s + 24-s − 4.51·25-s + 3.56·26-s − 27-s + 1.42·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.310·5-s + 0.408·6-s + 0.537·7-s − 0.353·8-s + 0.333·9-s + 0.219·10-s − 0.107·11-s − 0.288·12-s − 0.987·13-s − 0.380·14-s + 0.179·15-s + 0.250·16-s + 0.254·17-s − 0.235·18-s + 1.20·19-s − 0.155·20-s − 0.310·21-s + 0.0757·22-s + 0.208·23-s + 0.204·24-s − 0.903·25-s + 0.698·26-s − 0.192·27-s + 0.268·28-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: −1-1
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: 11
Selberg data: (2, 4002, ( :1/2), −1)(2,\ 4002,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+0.694T+5T2 1 + 0.694T + 5T^{2}
7 1−1.42T+7T2 1 - 1.42T + 7T^{2}
11 1+0.355T+11T2 1 + 0.355T + 11T^{2}
13 1+3.56T+13T2 1 + 3.56T + 13T^{2}
17 1−1.04T+17T2 1 - 1.04T + 17T^{2}
19 1−5.27T+19T2 1 - 5.27T + 19T^{2}
31 1+5.90T+31T2 1 + 5.90T + 31T^{2}
37 1+5.70T+37T2 1 + 5.70T + 37T^{2}
41 1−7.61T+41T2 1 - 7.61T + 41T^{2}
43 1+6.55T+43T2 1 + 6.55T + 43T^{2}
47 1−9.84T+47T2 1 - 9.84T + 47T^{2}
53 1−12.1T+53T2 1 - 12.1T + 53T^{2}
59 1+8.02T+59T2 1 + 8.02T + 59T^{2}
61 1+5.20T+61T2 1 + 5.20T + 61T^{2}
67 1−11.1T+67T2 1 - 11.1T + 67T^{2}
71 1+2.57T+71T2 1 + 2.57T + 71T^{2}
73 1+12.7T+73T2 1 + 12.7T + 73T^{2}
79 1−4.12T+79T2 1 - 4.12T + 79T^{2}
83 1+6.22T+83T2 1 + 6.22T + 83T^{2}
89 1+4.74T+89T2 1 + 4.74T + 89T^{2}
97 1+5.04T+97T2 1 + 5.04T + 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

−7.904387011286349671984957254510, −7.45009235220868819047093938540, −6.89592174540978437005166426180, −5.75066110639175680005303743730, −5.27548915948487723628126430947, −4.35158198556206738561240274824, −3.33976876639875764922556043736, −2.25355846588248715136308746074, −1.22301653701299104286698947725, 0, 1.22301653701299104286698947725, 2.25355846588248715136308746074, 3.33976876639875764922556043736, 4.35158198556206738561240274824, 5.27548915948487723628126430947, 5.75066110639175680005303743730, 6.89592174540978437005166426180, 7.45009235220868819047093938540, 7.904387011286349671984957254510

Graph of the ZZ-function along the critical line