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Properties

Label 2-616-1.1-c1-0-12
Degree 22
Conductor 616616
Sign −1-1
Analytic cond. 4.918784.91878
Root an. cond. 2.217832.21783
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  − 7-s − 3·9-s − 11-s − 6·13-s − 2·19-s + 4·23-s − 5·25-s − 2·29-s + 2·31-s + 2·37-s − 8·41-s − 2·47-s + 49-s − 10·53-s − 4·59-s + 10·61-s + 3·63-s + 4·67-s − 8·73-s + 77-s + 8·79-s + 9·81-s − 2·83-s − 6·89-s + 6·91-s + 2·97-s + 3·99-s + ⋯
L(s)  = 1  − 0.377·7-s − 9-s − 0.301·11-s − 1.66·13-s − 0.458·19-s + 0.834·23-s − 25-s − 0.371·29-s + 0.359·31-s + 0.328·37-s − 1.24·41-s − 0.291·47-s + 1/7·49-s − 1.37·53-s − 0.520·59-s + 1.28·61-s + 0.377·63-s + 0.488·67-s − 0.936·73-s + 0.113·77-s + 0.900·79-s + 81-s − 0.219·83-s − 0.635·89-s + 0.628·91-s + 0.203·97-s + 0.301·99-s + ⋯

Functional equation

Λ(s)=(616s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(616s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 616616    =    23⋅7⋅112^{3} \cdot 7 \cdot 11
Sign: −1-1
Analytic conductor: 4.918784.91878
Root analytic conductor: 2.217832.21783
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 616, ( :1/2), −1)(2,\ 616,\ (\ :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
7 1+T 1 + T
11 1+T 1 + T
good3 1+pT2 1 + p T^{2}
5 1+pT2 1 + p T^{2}
13 1+6T+pT2 1 + 6 T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1+2T+pT2 1 + 2 T + p T^{2}
23 1−4T+pT2 1 - 4 T + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
31 1−2T+pT2 1 - 2 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1+8T+pT2 1 + 8 T + p T^{2}
43 1+pT2 1 + p T^{2}
47 1+2T+pT2 1 + 2 T + p T^{2}
53 1+10T+pT2 1 + 10 T + p T^{2}
59 1+4T+pT2 1 + 4 T + p T^{2}
61 1−10T+pT2 1 - 10 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1+8T+pT2 1 + 8 T + p T^{2}
79 1−8T+pT2 1 - 8 T + p T^{2}
83 1+2T+pT2 1 + 2 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

−10.07995037027806350781622361938, −9.427364167513449399975184048294, −8.427472592413009369442981171559, −7.56186589309529403797122372766, −6.60336419837693223802202901848, −5.55808150974400203762583292939, −4.69735623090568767734453095829, −3.26677825137831447402592197381, −2.27748515390590495115985369379, 0, 2.27748515390590495115985369379, 3.26677825137831447402592197381, 4.69735623090568767734453095829, 5.55808150974400203762583292939, 6.60336419837693223802202901848, 7.56186589309529403797122372766, 8.427472592413009369442981171559, 9.427364167513449399975184048294, 10.07995037027806350781622361938

Graph of the ZZ-function along the critical line