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Properties

Label 2-1425-1.1-c3-0-42
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  + 3·2-s + 3·3-s + 4-s + 9·6-s − 32·7-s − 21·8-s + 9·9-s − 12·11-s + 3·12-s + 10·13-s − 96·14-s − 71·16-s + 30·17-s + 27·18-s + 19·19-s − 96·21-s − 36·22-s + 48·23-s − 63·24-s + 30·26-s + 27·27-s − 32·28-s + 150·29-s + 224·31-s − 45·32-s − 36·33-s + 90·34-s + ⋯
L(s)  = 1  + 1.06·2-s + 0.577·3-s + 1/8·4-s + 0.612·6-s − 1.72·7-s − 0.928·8-s + 1/3·9-s − 0.328·11-s + 0.0721·12-s + 0.213·13-s − 1.83·14-s − 1.10·16-s + 0.428·17-s + 0.353·18-s + 0.229·19-s − 0.997·21-s − 0.348·22-s + 0.435·23-s − 0.535·24-s + 0.226·26-s + 0.192·27-s − 0.215·28-s + 0.960·29-s + 1.29·31-s − 0.248·32-s − 0.189·33-s + 0.453·34-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 2.8969130822.896913082
L(12)L(\frac12) ≈\approx 2.8969130822.896913082
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 1−3T+p3T2 1 - 3 T + p^{3} T^{2}
7 1+32T+p3T2 1 + 32 T + p^{3} T^{2}
11 1+12T+p3T2 1 + 12 T + p^{3} T^{2}
13 1−10T+p3T2 1 - 10 T + p^{3} T^{2}
17 1−30T+p3T2 1 - 30 T + p^{3} T^{2}
23 1−48T+p3T2 1 - 48 T + p^{3} T^{2}
29 1−150T+p3T2 1 - 150 T + p^{3} T^{2}
31 1−224T+p3T2 1 - 224 T + p^{3} T^{2}
37 1+254T+p3T2 1 + 254 T + p^{3} T^{2}
41 1+54T+p3T2 1 + 54 T + p^{3} T^{2}
43 1−196T+p3T2 1 - 196 T + p^{3} T^{2}
47 1−504T+p3T2 1 - 504 T + p^{3} T^{2}
53 1+78T+p3T2 1 + 78 T + p^{3} T^{2}
59 1−132T+p3T2 1 - 132 T + p^{3} T^{2}
61 1−230T+p3T2 1 - 230 T + p^{3} T^{2}
67 1+740T+p3T2 1 + 740 T + p^{3} T^{2}
71 1+120T+p3T2 1 + 120 T + p^{3} T^{2}
73 1+122T+p3T2 1 + 122 T + p^{3} T^{2}
79 1−1184T+p3T2 1 - 1184 T + p^{3} T^{2}
83 1+612T+p3T2 1 + 612 T + p^{3} T^{2}
89 1−1050T+p3T2 1 - 1050 T + p^{3} T^{2}
97 1−1006T+p3T2 1 - 1006 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

−9.162400884711309089220915521174, −8.541042836419073513659124953867, −7.34875045895627760340977891363, −6.51313367719853467537336415887, −5.87619891811380768738933573538, −4.88154620560068261153022660998, −3.88991730760092177420279424584, −3.18378966244891502821089664892, −2.58629430980809380231631523352, −0.67522657057216800636212640141, 0.67522657057216800636212640141, 2.58629430980809380231631523352, 3.18378966244891502821089664892, 3.88991730760092177420279424584, 4.88154620560068261153022660998, 5.87619891811380768738933573538, 6.51313367719853467537336415887, 7.34875045895627760340977891363, 8.541042836419073513659124953867, 9.162400884711309089220915521174

Graph of the ZZ-function along the critical line