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Properties

Label 2-1425-1.1-c3-0-7
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3·3-s − 4·4-s − 6·6-s − 9·7-s − 24·8-s + 9·9-s − 62·11-s + 12·12-s + 38·13-s − 18·14-s − 16·16-s − 76·17-s + 18·18-s − 19·19-s + 27·21-s − 124·22-s − 42·23-s + 72·24-s + 76·26-s − 27·27-s + 36·28-s − 259·29-s − 120·31-s + 160·32-s + 186·33-s − 152·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 0.485·7-s − 1.06·8-s + 1/3·9-s − 1.69·11-s + 0.288·12-s + 0.810·13-s − 0.343·14-s − 1/4·16-s − 1.08·17-s + 0.235·18-s − 0.229·19-s + 0.280·21-s − 1.20·22-s − 0.380·23-s + 0.612·24-s + 0.573·26-s − 0.192·27-s + 0.242·28-s − 1.65·29-s − 0.695·31-s + 0.883·32-s + 0.981·33-s − 0.766·34-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: 00
Selberg data: (2, 1425, ( :3/2), 1)(2,\ 1425,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 0.57304085320.5730408532
L(12)L(\frac12) ≈\approx 0.57304085320.5730408532
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−pT+p3T2 1 - p T + p^{3} T^{2}
7 1+9T+p3T2 1 + 9 T + p^{3} T^{2}
11 1+62T+p3T2 1 + 62 T + p^{3} T^{2}
13 1−38T+p3T2 1 - 38 T + p^{3} T^{2}
17 1+76T+p3T2 1 + 76 T + p^{3} T^{2}
23 1+42T+p3T2 1 + 42 T + p^{3} T^{2}
29 1+259T+p3T2 1 + 259 T + p^{3} T^{2}
31 1+120T+p3T2 1 + 120 T + p^{3} T^{2}
37 1+230T+p3T2 1 + 230 T + p^{3} T^{2}
41 1−455T+p3T2 1 - 455 T + p^{3} T^{2}
43 1+340T+p3T2 1 + 340 T + p^{3} T^{2}
47 1−224T+p3T2 1 - 224 T + p^{3} T^{2}
53 1+61T+p3T2 1 + 61 T + p^{3} T^{2}
59 1+119T+p3T2 1 + 119 T + p^{3} T^{2}
61 1+113T+p3T2 1 + 113 T + p^{3} T^{2}
67 1−468T+p3T2 1 - 468 T + p^{3} T^{2}
71 1−995T+p3T2 1 - 995 T + p^{3} T^{2}
73 1+271T+p3T2 1 + 271 T + p^{3} T^{2}
79 1−318T+p3T2 1 - 318 T + p^{3} T^{2}
83 1+336T+p3T2 1 + 336 T + p^{3} T^{2}
89 1+945T+p3T2 1 + 945 T + p^{3} T^{2}
97 1+872T+p3T2 1 + 872 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

−9.200795026022793614075558243087, −8.409332455480086391465421255786, −7.47348356562674349305741938051, −6.42662326066186907476637869528, −5.69131685762520598111492908644, −5.09025681971480093379575772422, −4.14188933233335416567245987334, −3.31087857950875263797390348587, −2.12664951939184642648332641991, −0.32764376110764048234043855525, 0.32764376110764048234043855525, 2.12664951939184642648332641991, 3.31087857950875263797390348587, 4.14188933233335416567245987334, 5.09025681971480093379575772422, 5.69131685762520598111492908644, 6.42662326066186907476637869528, 7.47348356562674349305741938051, 8.409332455480086391465421255786, 9.200795026022793614075558243087

Graph of the ZZ-function along the critical line