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Properties

Label 2-50001-1.1-c1-0-0
Degree 22
Conductor 5000150001
Sign 11
Analytic cond. 399.259399.259
Root an. cond. 19.981419.9814
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  + 2·2-s + 3-s + 2·4-s − 2·5-s + 2·6-s − 7-s + 9-s − 4·10-s + 3·11-s + 2·12-s + 5·13-s − 2·14-s − 2·15-s − 4·16-s + 2·17-s + 2·18-s + 2·19-s − 4·20-s − 21-s + 6·22-s + 4·23-s − 25-s + 10·26-s + 27-s − 2·28-s − 8·29-s − 4·30-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s + 4-s − 0.894·5-s + 0.816·6-s − 0.377·7-s + 1/3·9-s − 1.26·10-s + 0.904·11-s + 0.577·12-s + 1.38·13-s − 0.534·14-s − 0.516·15-s − 16-s + 0.485·17-s + 0.471·18-s + 0.458·19-s − 0.894·20-s − 0.218·21-s + 1.27·22-s + 0.834·23-s − 1/5·25-s + 1.96·26-s + 0.192·27-s − 0.377·28-s − 1.48·29-s − 0.730·30-s + ⋯

Functional equation

Λ(s)=(50001s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 50001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(50001s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 50001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5000150001    =    3⋅7⋅23813 \cdot 7 \cdot 2381
Sign: 11
Analytic conductor: 399.259399.259
Root analytic conductor: 19.981419.9814
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 50001, ( :1/2), 1)(2,\ 50001,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 6.0779069416.077906941
L(12)L(\frac12) ≈\approx 6.0779069416.077906941
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)
bad3 1−T 1 - T
7 1+T 1 + T
2381 1+T 1 + T
good2 1−pT+pT2 1 - p T + p T^{2}
5 1+2T+pT2 1 + 2 T + p T^{2}
11 1−3T+pT2 1 - 3 T + p T^{2}
13 1−5T+pT2 1 - 5 T + p T^{2}
17 1−2T+pT2 1 - 2 T + p T^{2}
19 1−2T+pT2 1 - 2 T + p T^{2}
23 1−4T+pT2 1 - 4 T + p T^{2}
29 1+8T+pT2 1 + 8 T + p T^{2}
31 1+2T+pT2 1 + 2 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1+2T+pT2 1 + 2 T + p T^{2}
43 1−13T+pT2 1 - 13 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1−13T+pT2 1 - 13 T + p T^{2}
59 1−4T+pT2 1 - 4 T + p T^{2}
61 1+T+pT2 1 + T + p T^{2}
67 1+7T+pT2 1 + 7 T + p T^{2}
71 1−5T+pT2 1 - 5 T + p T^{2}
73 1−5T+pT2 1 - 5 T + p T^{2}
79 1+15T+pT2 1 + 15 T + p T^{2}
83 1−5T+pT2 1 - 5 T + p T^{2}
89 1−4T+pT2 1 - 4 T + p T^{2}
97 1+16T+pT2 1 + 16 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

−14.60698772776010, −13.95061966098781, −13.44174986831710, −13.19057987112790, −12.55861771968654, −12.05306265520921, −11.66724564420545, −11.02897594950457, −10.77566950754333, −9.650778536730069, −9.309739612502679, −8.738931333981395, −8.245433628403662, −7.354814743978140, −7.160015642777194, −6.403272082095717, −5.741874296736731, −5.469140411623893, −4.423827505047790, −3.998141202696205, −3.652972218813011, −3.205647960889482, −2.466157408987828, −1.527330675935686, −0.6798443057074887, 0.6798443057074887, 1.527330675935686, 2.466157408987828, 3.205647960889482, 3.652972218813011, 3.998141202696205, 4.423827505047790, 5.469140411623893, 5.741874296736731, 6.403272082095717, 7.160015642777194, 7.354814743978140, 8.245433628403662, 8.738931333981395, 9.309739612502679, 9.650778536730069, 10.77566950754333, 11.02897594950457, 11.66724564420545, 12.05306265520921, 12.55861771968654, 13.19057987112790, 13.44174986831710, 13.95061966098781, 14.60698772776010

Graph of the ZZ-function along the critical line