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Properties

Label 2-616-1.1-c1-0-13
Degree 22
Conductor 616616
Sign −1-1
Analytic cond. 4.918784.91878
Root an. cond. 2.217832.21783
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  + 1.56·3-s − 3.56·5-s − 7-s − 0.561·9-s + 11-s − 5.12·13-s − 5.56·15-s − 2·17-s + 3.12·19-s − 1.56·21-s − 5.56·23-s + 7.68·25-s − 5.56·27-s − 2·29-s − 6.43·31-s + 1.56·33-s + 3.56·35-s + 0.438·37-s − 8·39-s − 10·41-s + 4·43-s + 2·45-s + 10.2·47-s + 49-s − 3.12·51-s + 12.2·53-s − 3.56·55-s + ⋯
L(s)  = 1  + 0.901·3-s − 1.59·5-s − 0.377·7-s − 0.187·9-s + 0.301·11-s − 1.42·13-s − 1.43·15-s − 0.485·17-s + 0.716·19-s − 0.340·21-s − 1.15·23-s + 1.53·25-s − 1.07·27-s − 0.371·29-s − 1.15·31-s + 0.271·33-s + 0.602·35-s + 0.0720·37-s − 1.28·39-s − 1.56·41-s + 0.609·43-s + 0.298·45-s + 1.49·47-s + 0.142·49-s − 0.437·51-s + 1.68·53-s − 0.480·55-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: no
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 1−1.56T+3T2 1 - 1.56T + 3T^{2}
5 1+3.56T+5T2 1 + 3.56T + 5T^{2}
13 1+5.12T+13T2 1 + 5.12T + 13T^{2}
17 1+2T+17T2 1 + 2T + 17T^{2}
19 1−3.12T+19T2 1 - 3.12T + 19T^{2}
23 1+5.56T+23T2 1 + 5.56T + 23T^{2}
29 1+2T+29T2 1 + 2T + 29T^{2}
31 1+6.43T+31T2 1 + 6.43T + 31T^{2}
37 1−0.438T+37T2 1 - 0.438T + 37T^{2}
41 1+10T+41T2 1 + 10T + 41T^{2}
43 1−4T+43T2 1 - 4T + 43T^{2}
47 1−10.2T+47T2 1 - 10.2T + 47T^{2}
53 1−12.2T+53T2 1 - 12.2T + 53T^{2}
59 1+9.56T+59T2 1 + 9.56T + 59T^{2}
61 1−12.2T+61T2 1 - 12.2T + 61T^{2}
67 1−1.56T+67T2 1 - 1.56T + 67T^{2}
71 1+8.68T+71T2 1 + 8.68T + 71T^{2}
73 1−12.2T+73T2 1 - 12.2T + 73T^{2}
79 1+3.12T+79T2 1 + 3.12T + 79T^{2}
83 1+8T+83T2 1 + 8T + 83T^{2}
89 1+8.43T+89T2 1 + 8.43T + 89T^{2}
97 1−4.43T+97T2 1 - 4.43T + 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

−10.04900861894279974697769488082, −9.165298715982995101076461140387, −8.398420384640503098321040322054, −7.56165567820588145269391053470, −7.05477591295110836533108705515, −5.49624383309977074006990886382, −4.19955819997859575338975976910, −3.49576616605916737303806191153, −2.39887700497965227885451748768, 0, 2.39887700497965227885451748768, 3.49576616605916737303806191153, 4.19955819997859575338975976910, 5.49624383309977074006990886382, 7.05477591295110836533108705515, 7.56165567820588145269391053470, 8.398420384640503098321040322054, 9.165298715982995101076461140387, 10.04900861894279974697769488082

Graph of the ZZ-function along the critical line