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Properties

Label 2-6760-1.1-c1-0-131
Degree 22
Conductor 67606760
Sign −1-1
Analytic cond. 53.978853.9788
Root an. cond. 7.347037.34703
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 5-s + 9-s − 2·11-s − 2·15-s + 2·17-s − 2·19-s + 2·23-s + 25-s − 4·27-s − 6·29-s − 2·31-s − 4·33-s + 6·37-s − 2·41-s + 6·43-s − 45-s + 8·47-s − 7·49-s + 4·51-s − 2·53-s + 2·55-s − 4·57-s − 6·59-s − 14·61-s + 4·69-s − 10·71-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.447·5-s + 1/3·9-s − 0.603·11-s − 0.516·15-s + 0.485·17-s − 0.458·19-s + 0.417·23-s + 1/5·25-s − 0.769·27-s − 1.11·29-s − 0.359·31-s − 0.696·33-s + 0.986·37-s − 0.312·41-s + 0.914·43-s − 0.149·45-s + 1.16·47-s − 49-s + 0.560·51-s − 0.274·53-s + 0.269·55-s − 0.529·57-s − 0.781·59-s − 1.79·61-s + 0.481·69-s − 1.18·71-s + ⋯

Functional equation

Λ(s)=(6760s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 6760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(6760s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 67606760    =    23⋅5⋅1322^{3} \cdot 5 \cdot 13^{2}
Sign: −1-1
Analytic conductor: 53.978853.9788
Root analytic conductor: 7.347037.34703
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 6760, ( :1/2), −1)(2,\ 6760,\ (\ :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)
bad2 1 1
5 1+T 1 + T
13 1 1
good3 1−2T+pT2 1 - 2 T + p T^{2}
7 1+pT2 1 + p T^{2}
11 1+2T+pT2 1 + 2 T + p T^{2}
17 1−2T+pT2 1 - 2 T + p T^{2}
19 1+2T+pT2 1 + 2 T + p T^{2}
23 1−2T+pT2 1 - 2 T + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1+2T+pT2 1 + 2 T + p T^{2}
37 1−6T+pT2 1 - 6 T + p T^{2}
41 1+2T+pT2 1 + 2 T + p T^{2}
43 1−6T+pT2 1 - 6 T + p T^{2}
47 1−8T+pT2 1 - 8 T + p T^{2}
53 1+2T+pT2 1 + 2 T + p T^{2}
59 1+6T+pT2 1 + 6 T + p T^{2}
61 1+14T+pT2 1 + 14 T + p T^{2}
67 1+pT2 1 + p T^{2}
71 1+10T+pT2 1 + 10 T + p T^{2}
73 1−2T+pT2 1 - 2 T + p T^{2}
79 1+4T+pT2 1 + 4 T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1+2T+pT2 1 + 2 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

−7.65145326531316609749473144008, −7.33387504597643637570519717846, −6.20297819498089116422846262337, −5.52292241535810330877601261053, −4.57569787487438471292926229119, −3.85287056779756280024412088003, −3.08843563443615203217480631996, −2.50392315128847178535779020476, −1.48618888866608499865943758358, 0, 1.48618888866608499865943758358, 2.50392315128847178535779020476, 3.08843563443615203217480631996, 3.85287056779756280024412088003, 4.57569787487438471292926229119, 5.52292241535810330877601261053, 6.20297819498089116422846262337, 7.33387504597643637570519717846, 7.65145326531316609749473144008

Graph of the ZZ-function along the critical line