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Properties

Label 2-4002-1.1-c1-0-42
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 − 4.40·5-s − 6-s − 0.551·7-s − 8-s + 9-s + 4.40·10-s + 1.47·11-s + 12-s − 2.62·13-s + 0.551·14-s − 4.40·15-s + 16-s − 1.48·17-s − 18-s + 2.46·19-s − 4.40·20-s − 0.551·21-s − 1.47·22-s − 23-s − 24-s + 14.4·25-s + 2.62·26-s + 27-s − 0.551·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.97·5-s − 0.408·6-s − 0.208·7-s − 0.353·8-s + 0.333·9-s + 1.39·10-s + 0.444·11-s + 0.288·12-s − 0.727·13-s + 0.147·14-s − 1.13·15-s + 0.250·16-s − 0.360·17-s − 0.235·18-s + 0.566·19-s − 0.986·20-s − 0.120·21-s − 0.314·22-s − 0.208·23-s − 0.204·24-s + 2.88·25-s + 0.514·26-s + 0.192·27-s − 0.104·28-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+4.40T+5T2 1 + 4.40T + 5T^{2}
7 1+0.551T+7T2 1 + 0.551T + 7T^{2}
11 1−1.47T+11T2 1 - 1.47T + 11T^{2}
13 1+2.62T+13T2 1 + 2.62T + 13T^{2}
17 1+1.48T+17T2 1 + 1.48T + 17T^{2}
19 1−2.46T+19T2 1 - 2.46T + 19T^{2}
31 1−5.87T+31T2 1 - 5.87T + 31T^{2}
37 1−4.22T+37T2 1 - 4.22T + 37T^{2}
41 1+0.176T+41T2 1 + 0.176T + 41T^{2}
43 1−11.2T+43T2 1 - 11.2T + 43T^{2}
47 1+3.46T+47T2 1 + 3.46T + 47T^{2}
53 1+3.90T+53T2 1 + 3.90T + 53T^{2}
59 1−9.14T+59T2 1 - 9.14T + 59T^{2}
61 1+11.4T+61T2 1 + 11.4T + 61T^{2}
67 1+13.9T+67T2 1 + 13.9T + 67T^{2}
71 1+15.0T+71T2 1 + 15.0T + 71T^{2}
73 1−6.61T+73T2 1 - 6.61T + 73T^{2}
79 1+1.53T+79T2 1 + 1.53T + 79T^{2}
83 1+5.66T+83T2 1 + 5.66T + 83T^{2}
89 1+6.00T+89T2 1 + 6.00T + 89T^{2}
97 1+10.1T+97T2 1 + 10.1T + 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

−8.033425280244344871730167380733, −7.53429302683910003198798419067, −7.03064216557184640527348051845, −6.15614789561547454589583403927, −4.73397804536037602947924394714, −4.20492927554533029888724695857, −3.27663752488207807709485966493, −2.65462199985882722696557368439, −1.15136100559167426286202914305, 0, 1.15136100559167426286202914305, 2.65462199985882722696557368439, 3.27663752488207807709485966493, 4.20492927554533029888724695857, 4.73397804536037602947924394714, 6.15614789561547454589583403927, 7.03064216557184640527348051845, 7.53429302683910003198798419067, 8.033425280244344871730167380733

Graph of the ZZ-function along the critical line