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Properties

Label 2-825-1.1-c1-0-29
Degree 22
Conductor 825825
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.48·2-s − 3-s + 0.193·4-s − 1.48·6-s − 1.19·7-s − 2.67·8-s + 9-s + 11-s − 0.193·12-s − 0.806·13-s − 1.76·14-s − 4.35·16-s − 3.76·17-s + 1.48·18-s − 5.35·19-s + 1.19·21-s + 1.48·22-s − 4·23-s + 2.67·24-s − 1.19·26-s − 27-s − 0.231·28-s − 4.31·29-s + 0.962·31-s − 1.09·32-s − 33-s − 5.58·34-s + ⋯
L(s)  = 1  + 1.04·2-s − 0.577·3-s + 0.0969·4-s − 0.604·6-s − 0.451·7-s − 0.945·8-s + 0.333·9-s + 0.301·11-s − 0.0559·12-s − 0.223·13-s − 0.472·14-s − 1.08·16-s − 0.913·17-s + 0.349·18-s − 1.22·19-s + 0.260·21-s + 0.315·22-s − 0.834·23-s + 0.546·24-s − 0.234·26-s − 0.192·27-s − 0.0437·28-s − 0.800·29-s + 0.172·31-s − 0.193·32-s − 0.174·33-s − 0.957·34-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: −1-1
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: 11
Selberg data: (2, 825, ( :1/2), −1)(2,\ 825,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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 1−1.48T+2T2 1 - 1.48T + 2T^{2}
7 1+1.19T+7T2 1 + 1.19T + 7T^{2}
13 1+0.806T+13T2 1 + 0.806T + 13T^{2}
17 1+3.76T+17T2 1 + 3.76T + 17T^{2}
19 1+5.35T+19T2 1 + 5.35T + 19T^{2}
23 1+4T+23T2 1 + 4T + 23T^{2}
29 1+4.31T+29T2 1 + 4.31T + 29T^{2}
31 1−0.962T+31T2 1 - 0.962T + 31T^{2}
37 1+1.61T+37T2 1 + 1.61T + 37T^{2}
41 1−9.08T+41T2 1 - 9.08T + 41T^{2}
43 1+4.41T+43T2 1 + 4.41T + 43T^{2}
47 1+12.3T+47T2 1 + 12.3T + 47T^{2}
53 1+1.42T+53T2 1 + 1.42T + 53T^{2}
59 1−13.2T+59T2 1 - 13.2T + 59T^{2}
61 1+0.0752T+61T2 1 + 0.0752T + 61T^{2}
67 1−2.70T+67T2 1 - 2.70T + 67T^{2}
71 1+14.0T+71T2 1 + 14.0T + 71T^{2}
73 1−10.7T+73T2 1 - 10.7T + 73T^{2}
79 1−13.9T+79T2 1 - 13.9T + 79T^{2}
83 1−9.89T+83T2 1 - 9.89T + 83T^{2}
89 1−16.8T+89T2 1 - 16.8T + 89T^{2}
97 1+11.4T+97T2 1 + 11.4T + 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

−9.824727548796061474124841653805, −9.083657907431766356766631054875, −8.068013884955445269719119145717, −6.66540924706850267965456337882, −6.28062773744403945913256246318, −5.24767170607294390761269817532, −4.38568330778879800444946357376, −3.61582496401322644909572348923, −2.22506912740385525617645534196, 0, 2.22506912740385525617645534196, 3.61582496401322644909572348923, 4.38568330778879800444946357376, 5.24767170607294390761269817532, 6.28062773744403945913256246318, 6.66540924706850267965456337882, 8.068013884955445269719119145717, 9.083657907431766356766631054875, 9.824727548796061474124841653805

Graph of the ZZ-function along the critical line