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Properties

Label 2-1425-1.1-c3-0-160
Degree 22
Conductor 14251425
Sign 11
Analytic cond. 84.077784.0777
Root an. cond. 9.169399.16939
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 22

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3·3-s − 7·4-s + 3·6-s + 4·7-s + 15·8-s + 9·9-s − 68·11-s + 21·12-s − 82·13-s − 4·14-s + 41·16-s − 86·17-s − 9·18-s + 19·19-s − 12·21-s + 68·22-s + 18·23-s − 45·24-s + 82·26-s − 27·27-s − 28·28-s + 30·29-s − 298·31-s − 161·32-s + 204·33-s + 86·34-s + ⋯
L(s)  = 1  − 0.353·2-s − 0.577·3-s − 7/8·4-s + 0.204·6-s + 0.215·7-s + 0.662·8-s + 1/3·9-s − 1.86·11-s + 0.505·12-s − 1.74·13-s − 0.0763·14-s + 0.640·16-s − 1.22·17-s − 0.117·18-s + 0.229·19-s − 0.124·21-s + 0.658·22-s + 0.163·23-s − 0.382·24-s + 0.618·26-s − 0.192·27-s − 0.188·28-s + 0.192·29-s − 1.72·31-s − 0.889·32-s + 1.07·33-s + 0.433·34-s + ⋯

Functional equation

Λ(s)=(1425s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1425s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14251425    =    3⋅52⋅193 \cdot 5^{2} \cdot 19
Sign: 11
Analytic conductor: 84.077784.0777
Root analytic conductor: 9.169399.16939
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 22
Selberg data: (2, 1425, ( :3/2), 1)(2,\ 1425,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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+pT 1 + p T
5 1 1
19 1−pT 1 - p T
good2 1+T+p3T2 1 + T + p^{3} T^{2}
7 1−4T+p3T2 1 - 4 T + p^{3} T^{2}
11 1+68T+p3T2 1 + 68 T + p^{3} T^{2}
13 1+82T+p3T2 1 + 82 T + p^{3} T^{2}
17 1+86T+p3T2 1 + 86 T + p^{3} T^{2}
23 1−18T+p3T2 1 - 18 T + p^{3} T^{2}
29 1−30T+p3T2 1 - 30 T + p^{3} T^{2}
31 1+298T+p3T2 1 + 298 T + p^{3} T^{2}
37 1−34T+p3T2 1 - 34 T + p^{3} T^{2}
41 1−52T+p3T2 1 - 52 T + p^{3} T^{2}
43 1+482T+p3T2 1 + 482 T + p^{3} T^{2}
47 1−114T+p3T2 1 - 114 T + p^{3} T^{2}
53 1+362T+p3T2 1 + 362 T + p^{3} T^{2}
59 1+210T+p3T2 1 + 210 T + p^{3} T^{2}
61 1+718T+p3T2 1 + 718 T + p^{3} T^{2}
67 1−904T+p3T2 1 - 904 T + p^{3} T^{2}
71 1+988T+p3T2 1 + 988 T + p^{3} T^{2}
73 1−488T+p3T2 1 - 488 T + p^{3} T^{2}
79 1+530T+p3T2 1 + 530 T + p^{3} T^{2}
83 1+1032T+p3T2 1 + 1032 T + p^{3} T^{2}
89 1+880T+p3T2 1 + 880 T + p^{3} T^{2}
97 1+246T+p3T2 1 + 246 T + p^{3} 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

−8.303904225435382578141588312731, −7.63658211570533949103562651216, −6.95737333029022808985019818402, −5.55685281590411326916815134504, −4.99920887032309436520352025308, −4.43597035256288652302310283057, −2.95275784224818347832220648653, −1.82295564005590077081690964400, 0, 0, 1.82295564005590077081690964400, 2.95275784224818347832220648653, 4.43597035256288652302310283057, 4.99920887032309436520352025308, 5.55685281590411326916815134504, 6.95737333029022808985019818402, 7.63658211570533949103562651216, 8.303904225435382578141588312731

Graph of the ZZ-function along the critical line