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Properties

Label 2-4002-1.1-c1-0-29
Degree 22
Conductor 40024002
Sign 11
Analytic cond. 31.956131.9561
Root an. cond. 5.652975.65297
Motivic weight 11
Arithmetic yes
Rational yes
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 − 5-s − 6-s − 8-s + 9-s + 10-s + 5·11-s + 12-s + 3·13-s − 15-s + 16-s + 4·17-s − 18-s + 4·19-s − 20-s − 5·22-s − 23-s − 24-s − 4·25-s − 3·26-s + 27-s − 29-s + 30-s + 5·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.50·11-s + 0.288·12-s + 0.832·13-s − 0.258·15-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 0.917·19-s − 0.223·20-s − 1.06·22-s − 0.208·23-s − 0.204·24-s − 4/5·25-s − 0.588·26-s + 0.192·27-s − 0.185·29-s + 0.182·30-s + 0.898·31-s − 0.176·32-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: yes
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 1.9584394221.958439422
L(12)L(\frac12) ≈\approx 1.9584394221.958439422
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+T+pT2 1 + T + p T^{2}
7 1+pT2 1 + p T^{2}
11 1−5T+pT2 1 - 5 T + p T^{2}
13 1−3T+pT2 1 - 3 T + p T^{2}
17 1−4T+pT2 1 - 4 T + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
31 1−5T+pT2 1 - 5 T + p T^{2}
37 1+3T+pT2 1 + 3 T + p T^{2}
41 1−9T+pT2 1 - 9 T + p T^{2}
43 1+6T+pT2 1 + 6 T + p T^{2}
47 1−10T+pT2 1 - 10 T + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1+7T+pT2 1 + 7 T + p T^{2}
61 1−T+pT2 1 - T + p T^{2}
67 1+13T+pT2 1 + 13 T + p T^{2}
71 1+5T+pT2 1 + 5 T + p T^{2}
73 1+4T+pT2 1 + 4 T + p T^{2}
79 1+2T+pT2 1 + 2 T + p T^{2}
83 1+6T+pT2 1 + 6 T + p T^{2}
89 1+10T+pT2 1 + 10 T + p T^{2}
97 1−18T+pT2 1 - 18 T + p T^{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

−8.582903535103051671220586523393, −7.70107674785881678622186592031, −7.31715568476819843350858462741, −6.32891082747463643586427384833, −5.74985936346118792546956940632, −4.41955874685816877821382920981, −3.67659661269898197194758563403, −3.03538156105061200759572949400, −1.70965449751274279349505212042, −0.946094456404816744025198268194, 0.946094456404816744025198268194, 1.70965449751274279349505212042, 3.03538156105061200759572949400, 3.67659661269898197194758563403, 4.41955874685816877821382920981, 5.74985936346118792546956940632, 6.32891082747463643586427384833, 7.31715568476819843350858462741, 7.70107674785881678622186592031, 8.582903535103051671220586523393

Graph of the ZZ-function along the critical line