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Properties

Label 2-616-1.1-c1-0-8
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  − 2.56·3-s + 0.561·5-s − 7-s + 3.56·9-s + 11-s + 3.12·13-s − 1.43·15-s − 2·17-s − 5.12·19-s + 2.56·21-s − 1.43·23-s − 4.68·25-s − 1.43·27-s − 2·29-s − 10.5·31-s − 2.56·33-s − 0.561·35-s + 4.56·37-s − 8·39-s − 10·41-s + 4·43-s + 2.00·45-s − 6.24·47-s + 49-s + 5.12·51-s − 4.24·53-s + 0.561·55-s + ⋯
L(s)  = 1  − 1.47·3-s + 0.251·5-s − 0.377·7-s + 1.18·9-s + 0.301·11-s + 0.866·13-s − 0.371·15-s − 0.485·17-s − 1.17·19-s + 0.558·21-s − 0.299·23-s − 0.936·25-s − 0.276·27-s − 0.371·29-s − 1.89·31-s − 0.445·33-s − 0.0949·35-s + 0.749·37-s − 1.28·39-s − 1.56·41-s + 0.609·43-s + 0.298·45-s − 0.911·47-s + 0.142·49-s + 0.717·51-s − 0.583·53-s + 0.0757·55-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+2.56T+3T2 1 + 2.56T + 3T^{2}
5 1−0.561T+5T2 1 - 0.561T + 5T^{2}
13 1−3.12T+13T2 1 - 3.12T + 13T^{2}
17 1+2T+17T2 1 + 2T + 17T^{2}
19 1+5.12T+19T2 1 + 5.12T + 19T^{2}
23 1+1.43T+23T2 1 + 1.43T + 23T^{2}
29 1+2T+29T2 1 + 2T + 29T^{2}
31 1+10.5T+31T2 1 + 10.5T + 31T^{2}
37 1−4.56T+37T2 1 - 4.56T + 37T^{2}
41 1+10T+41T2 1 + 10T + 41T^{2}
43 1−4T+43T2 1 - 4T + 43T^{2}
47 1+6.24T+47T2 1 + 6.24T + 47T^{2}
53 1+4.24T+53T2 1 + 4.24T + 53T^{2}
59 1+5.43T+59T2 1 + 5.43T + 59T^{2}
61 1+4.24T+61T2 1 + 4.24T + 61T^{2}
67 1+2.56T+67T2 1 + 2.56T + 67T^{2}
71 1−3.68T+71T2 1 - 3.68T + 71T^{2}
73 1+4.24T+73T2 1 + 4.24T + 73T^{2}
79 1−5.12T+79T2 1 - 5.12T + 79T^{2}
83 1+8T+83T2 1 + 8T + 83T^{2}
89 1+12.5T+89T2 1 + 12.5T + 89T^{2}
97 1−8.56T+97T2 1 - 8.56T + 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

−10.43913578322948856267191085054, −9.484333193830727732894361336105, −8.527295296177033558571543689271, −7.23420393626824865651930951731, −6.24770212755611380422417529504, −5.89832044716397750489797354287, −4.72563752578301480791720225628, −3.67537811329081465669048290880, −1.77691171676713607451976111874, 0, 1.77691171676713607451976111874, 3.67537811329081465669048290880, 4.72563752578301480791720225628, 5.89832044716397750489797354287, 6.24770212755611380422417529504, 7.23420393626824865651930951731, 8.527295296177033558571543689271, 9.484333193830727732894361336105, 10.43913578322948856267191085054

Graph of the ZZ-function along the critical line