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Properties

Label 2-825-1.1-c1-0-15
Degree 22
Conductor 825825
Sign 11
Analytic cond. 6.587656.58765
Root an. cond. 2.566642.56664
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.36·2-s + 3-s − 0.141·4-s + 1.36·6-s + 2.50·7-s − 2.91·8-s + 9-s + 11-s − 0.141·12-s + 1.14·13-s + 3.41·14-s − 3.69·16-s + 7.64·17-s + 1.36·18-s + 1.77·19-s + 2.50·21-s + 1.36·22-s + 1.41·23-s − 2.91·24-s + 1.55·26-s + 27-s − 0.353·28-s − 0.726·29-s + 2.85·31-s + 0.797·32-s + 33-s + 10.4·34-s + ⋯
L(s)  = 1  + 0.964·2-s + 0.577·3-s − 0.0706·4-s + 0.556·6-s + 0.946·7-s − 1.03·8-s + 0.333·9-s + 0.301·11-s − 0.0408·12-s + 0.316·13-s + 0.912·14-s − 0.924·16-s + 1.85·17-s + 0.321·18-s + 0.407·19-s + 0.546·21-s + 0.290·22-s + 0.294·23-s − 0.595·24-s + 0.305·26-s + 0.192·27-s − 0.0668·28-s − 0.134·29-s + 0.513·31-s + 0.141·32-s + 0.174·33-s + 1.78·34-s + ⋯

Functional equation

Λ(s)=(825s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(825s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 825825    =    3⋅52⋅113 \cdot 5^{2} \cdot 11
Sign: 11
Analytic conductor: 6.587656.58765
Root analytic conductor: 2.566642.56664
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 825, ( :1/2), 1)(2,\ 825,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.0503684873.050368487
L(12)L(\frac12) ≈\approx 3.0503684873.050368487
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
11 1−T 1 - T
good2 1−1.36T+2T2 1 - 1.36T + 2T^{2}
7 1−2.50T+7T2 1 - 2.50T + 7T^{2}
13 1−1.14T+13T2 1 - 1.14T + 13T^{2}
17 1−7.64T+17T2 1 - 7.64T + 17T^{2}
19 1−1.77T+19T2 1 - 1.77T + 19T^{2}
23 1−1.41T+23T2 1 - 1.41T + 23T^{2}
29 1+0.726T+29T2 1 + 0.726T + 29T^{2}
31 1−2.85T+31T2 1 - 2.85T + 31T^{2}
37 1+8.42T+37T2 1 + 8.42T + 37T^{2}
41 1−0.636T+41T2 1 - 0.636T + 41T^{2}
43 1+12.6T+43T2 1 + 12.6T + 43T^{2}
47 1−6.14T+47T2 1 - 6.14T + 47T^{2}
53 1+12.0T+53T2 1 + 12.0T + 53T^{2}
59 1+3.41T+59T2 1 + 3.41T + 59T^{2}
61 1−4.59T+61T2 1 - 4.59T + 61T^{2}
67 1+9.32T+67T2 1 + 9.32T + 67T^{2}
71 1−5.85T+71T2 1 - 5.85T + 71T^{2}
73 1−7.55T+73T2 1 - 7.55T + 73T^{2}
79 1−6.91T+79T2 1 - 6.91T + 79T^{2}
83 1+6.17T+83T2 1 + 6.17T + 83T^{2}
89 1−3.45T+89T2 1 - 3.45T + 89T^{2}
97 1+19.4T+97T2 1 + 19.4T + 97T^{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

−10.15374733122215738704046531845, −9.348649416063764158760393437178, −8.430644717312248673885648608673, −7.79553841138608589125578986841, −6.63666534237307394030041054888, −5.48954106279152338801918672356, −4.87809759700423328069375888932, −3.77371419785787475240012480777, −3.04866917193860400925149613291, −1.45428727031405064443889610082, 1.45428727031405064443889610082, 3.04866917193860400925149613291, 3.77371419785787475240012480777, 4.87809759700423328069375888932, 5.48954106279152338801918672356, 6.63666534237307394030041054888, 7.79553841138608589125578986841, 8.430644717312248673885648608673, 9.348649416063764158760393437178, 10.15374733122215738704046531845

Graph of the ZZ-function along the critical line