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Properties

Label 2-4650-1.1-c1-0-23
Degree 22
Conductor 46504650
Sign 11
Analytic cond. 37.130437.1304
Root an. cond. 6.093476.09347
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 + 6-s − 5·7-s + 8-s + 9-s − 11-s + 12-s − 5·14-s + 16-s + 4·17-s + 18-s + 3·19-s − 5·21-s − 22-s + 23-s + 24-s + 27-s − 5·28-s + 6·29-s − 31-s + 32-s − 33-s + 4·34-s + 36-s + 4·37-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.88·7-s + 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s − 1.33·14-s + 1/4·16-s + 0.970·17-s + 0.235·18-s + 0.688·19-s − 1.09·21-s − 0.213·22-s + 0.208·23-s + 0.204·24-s + 0.192·27-s − 0.944·28-s + 1.11·29-s − 0.179·31-s + 0.176·32-s − 0.174·33-s + 0.685·34-s + 1/6·36-s + 0.657·37-s + ⋯

Functional equation

Λ(s)=(4650s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4650s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 46504650    =    2⋅3⋅52⋅312 \cdot 3 \cdot 5^{2} \cdot 31
Sign: 11
Analytic conductor: 37.130437.1304
Root analytic conductor: 6.093476.09347
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4650, ( :1/2), 1)(2,\ 4650,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.1344781563.134478156
L(12)L(\frac12) ≈\approx 3.1344781563.134478156
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
5 1 1
31 1+T 1 + T
good7 1+5T+pT2 1 + 5 T + p T^{2}
11 1+T+pT2 1 + T + p T^{2}
13 1+pT2 1 + p T^{2}
17 1−4T+pT2 1 - 4 T + p T^{2}
19 1−3T+pT2 1 - 3 T + p T^{2}
23 1−T+pT2 1 - T + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
37 1−4T+pT2 1 - 4 T + p T^{2}
41 1−2T+pT2 1 - 2 T + p T^{2}
43 1−T+pT2 1 - T + p T^{2}
47 1+4T+pT2 1 + 4 T + p T^{2}
53 1+3T+pT2 1 + 3 T + p T^{2}
59 1+14T+pT2 1 + 14 T + p T^{2}
61 1−14T+pT2 1 - 14 T + p T^{2}
67 1+10T+pT2 1 + 10 T + p T^{2}
71 1−9T+pT2 1 - 9 T + p T^{2}
73 1−7T+pT2 1 - 7 T + p T^{2}
79 1−15T+pT2 1 - 15 T + p T^{2}
83 1−10T+pT2 1 - 10 T + p T^{2}
89 1+T+pT2 1 + T + p T^{2}
97 1+10T+pT2 1 + 10 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

−8.161945058453112513595584410005, −7.49391269460864216446385515019, −6.74077270636893430153068344267, −6.17721768161906194996196903416, −5.41775919238442297270092743082, −4.48797396004496236766001778692, −3.42468065614581908043689633451, −3.20325720671657470667140663817, −2.33091504521382109584331939466, −0.850626932479884459655088516509, 0.850626932479884459655088516509, 2.33091504521382109584331939466, 3.20325720671657470667140663817, 3.42468065614581908043689633451, 4.48797396004496236766001778692, 5.41775919238442297270092743082, 6.17721768161906194996196903416, 6.74077270636893430153068344267, 7.49391269460864216446385515019, 8.161945058453112513595584410005

Graph of the ZZ-function along the critical line