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Properties

Label 2-1425-1.1-c1-0-20
Degree 22
Conductor 14251425
Sign 11
Analytic cond. 11.378611.3786
Root an. cond. 3.373233.37323
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 + 4·7-s + 3·8-s + 9-s + 3·11-s − 12-s − 4·14-s − 16-s − 18-s + 19-s + 4·21-s − 3·22-s + 23-s + 3·24-s + 27-s − 4·28-s − 5·29-s + 9·31-s − 5·32-s + 3·33-s − 36-s − 2·37-s − 38-s + 6·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s + 1.51·7-s + 1.06·8-s + 1/3·9-s + 0.904·11-s − 0.288·12-s − 1.06·14-s − 1/4·16-s − 0.235·18-s + 0.229·19-s + 0.872·21-s − 0.639·22-s + 0.208·23-s + 0.612·24-s + 0.192·27-s − 0.755·28-s − 0.928·29-s + 1.61·31-s − 0.883·32-s + 0.522·33-s − 1/6·36-s − 0.328·37-s − 0.162·38-s + 0.937·41-s + ⋯

Functional equation

Λ(s)=(1425s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1425s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, 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: 11.378611.3786
Root analytic conductor: 3.373233.37323
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1425, ( :1/2), 1)(2,\ 1425,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.6131971531.613197153
L(12)L(\frac12) ≈\approx 1.6131971531.613197153
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
5 1 1
19 1−T 1 - T
good2 1+T+pT2 1 + T + p T^{2}
7 1−4T+pT2 1 - 4 T + p T^{2}
11 1−3T+pT2 1 - 3 T + p T^{2}
13 1+pT2 1 + p T^{2}
17 1+pT2 1 + p T^{2}
23 1−T+pT2 1 - T + p T^{2}
29 1+5T+pT2 1 + 5 T + p T^{2}
31 1−9T+pT2 1 - 9 T + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1+8T+pT2 1 + 8 T + p T^{2}
47 1+8T+pT2 1 + 8 T + p T^{2}
53 1+T+pT2 1 + T + p T^{2}
59 1+12T+pT2 1 + 12 T + p T^{2}
61 1−5T+pT2 1 - 5 T + p T^{2}
67 1+5T+pT2 1 + 5 T + p T^{2}
71 1−10T+pT2 1 - 10 T + p T^{2}
73 1−11T+pT2 1 - 11 T + p T^{2}
79 1−11T+pT2 1 - 11 T + p T^{2}
83 1+9T+pT2 1 + 9 T + p T^{2}
89 1−15T+pT2 1 - 15 T + p T^{2}
97 1−18T+pT2 1 - 18 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

−9.375586930532197171435102084043, −8.721157197040644373439868435028, −8.059132773379975443514972561189, −7.56262888744563490820286573499, −6.46872919351205495550038689249, −5.09570305472347318169914513335, −4.52175123062198989941702185993, −3.54975381370698875378500028111, −1.98349098766325641228995661317, −1.11188483392532769774032160005, 1.11188483392532769774032160005, 1.98349098766325641228995661317, 3.54975381370698875378500028111, 4.52175123062198989941702185993, 5.09570305472347318169914513335, 6.46872919351205495550038689249, 7.56262888744563490820286573499, 8.059132773379975443514972561189, 8.721157197040644373439868435028, 9.375586930532197171435102084043

Graph of the ZZ-function along the critical line