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Properties

Label 2-459-1.1-c1-0-14
Degree 22
Conductor 459459
Sign −1-1
Analytic cond. 3.665133.66513
Root an. cond. 1.914451.91445
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s + 5-s − 2·7-s + 3·8-s − 10-s − 5·13-s + 2·14-s − 16-s + 17-s − 19-s − 20-s + 23-s − 4·25-s + 5·26-s + 2·28-s − 9·29-s − 8·31-s − 5·32-s − 34-s − 2·35-s − 2·37-s + 38-s + 3·40-s + 3·41-s + 7·43-s − 46-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.447·5-s − 0.755·7-s + 1.06·8-s − 0.316·10-s − 1.38·13-s + 0.534·14-s − 1/4·16-s + 0.242·17-s − 0.229·19-s − 0.223·20-s + 0.208·23-s − 4/5·25-s + 0.980·26-s + 0.377·28-s − 1.67·29-s − 1.43·31-s − 0.883·32-s − 0.171·34-s − 0.338·35-s − 0.328·37-s + 0.162·38-s + 0.474·40-s + 0.468·41-s + 1.06·43-s − 0.147·46-s + ⋯

Functional equation

Λ(s)=(459s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(459s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 459459    =    33⋅173^{3} \cdot 17
Sign: −1-1
Analytic conductor: 3.665133.66513
Root analytic conductor: 1.914451.91445
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 459, ( :1/2), −1)(2,\ 459,\ (\ :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)
bad3 1 1
17 1−T 1 - T
good2 1+T+pT2 1 + T + p T^{2}
5 1−T+pT2 1 - T + p T^{2}
7 1+2T+pT2 1 + 2 T + p T^{2}
11 1+pT2 1 + p T^{2}
13 1+5T+pT2 1 + 5 T + p T^{2}
19 1+T+pT2 1 + T + p T^{2}
23 1−T+pT2 1 - T + p T^{2}
29 1+9T+pT2 1 + 9 T + p T^{2}
31 1+8T+pT2 1 + 8 T + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1−3T+pT2 1 - 3 T + p T^{2}
43 1−7T+pT2 1 - 7 T + p T^{2}
47 1+6T+pT2 1 + 6 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1+10T+pT2 1 + 10 T + p T^{2}
67 1−T+pT2 1 - T + p T^{2}
71 1−11T+pT2 1 - 11 T + p T^{2}
73 1−6T+pT2 1 - 6 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 1+4T+pT2 1 + 4 T + p T^{2}
89 1+2T+pT2 1 + 2 T + p T^{2}
97 1−2T+pT2 1 - 2 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

−10.30729061604743667613050680491, −9.486856844687813433475551610921, −9.232263132127470389713380311071, −7.85873665118377503458386090870, −7.19141577818664657143012339932, −5.87858091946480144424247943451, −4.87958568118636028443679374257, −3.59857441008458632308323977451, −1.99713451755220112161339413052, 0, 1.99713451755220112161339413052, 3.59857441008458632308323977451, 4.87958568118636028443679374257, 5.87858091946480144424247943451, 7.19141577818664657143012339932, 7.85873665118377503458386090870, 9.232263132127470389713380311071, 9.486856844687813433475551610921, 10.30729061604743667613050680491

Graph of the ZZ-function along the critical line