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Properties

Label 2-4368-1.1-c1-0-9
Degree 22
Conductor 43684368
Sign 11
Analytic cond. 34.878634.8786
Root an. cond. 5.905815.90581
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 0.561·5-s + 7-s + 9-s − 2.56·11-s + 13-s + 0.561·15-s + 5.68·17-s − 7.68·19-s − 21-s + 1.43·23-s − 4.68·25-s − 27-s − 5.68·29-s + 10.2·31-s + 2.56·33-s − 0.561·35-s − 3.43·37-s − 39-s + 7.12·41-s + 10.5·43-s − 0.561·45-s + 49-s − 5.68·51-s − 4.24·53-s + 1.43·55-s + 7.68·57-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.251·5-s + 0.377·7-s + 0.333·9-s − 0.772·11-s + 0.277·13-s + 0.144·15-s + 1.37·17-s − 1.76·19-s − 0.218·21-s + 0.299·23-s − 0.936·25-s − 0.192·27-s − 1.05·29-s + 1.84·31-s + 0.445·33-s − 0.0949·35-s − 0.565·37-s − 0.160·39-s + 1.11·41-s + 1.61·43-s − 0.0837·45-s + 0.142·49-s − 0.796·51-s − 0.583·53-s + 0.193·55-s + 1.01·57-s + ⋯

Functional equation

Λ(s)=(4368s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4368s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 43684368    =    24⋅3⋅7⋅132^{4} \cdot 3 \cdot 7 \cdot 13
Sign: 11
Analytic conductor: 34.878634.8786
Root analytic conductor: 5.905815.90581
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4368, ( :1/2), 1)(2,\ 4368,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.3021825891.302182589
L(12)L(\frac12) ≈\approx 1.3021825891.302182589
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
3 1+T 1 + T
7 1−T 1 - T
13 1−T 1 - T
good5 1+0.561T+5T2 1 + 0.561T + 5T^{2}
11 1+2.56T+11T2 1 + 2.56T + 11T^{2}
17 1−5.68T+17T2 1 - 5.68T + 17T^{2}
19 1+7.68T+19T2 1 + 7.68T + 19T^{2}
23 1−1.43T+23T2 1 - 1.43T + 23T^{2}
29 1+5.68T+29T2 1 + 5.68T + 29T^{2}
31 1−10.2T+31T2 1 - 10.2T + 31T^{2}
37 1+3.43T+37T2 1 + 3.43T + 37T^{2}
41 1−7.12T+41T2 1 - 7.12T + 41T^{2}
43 1−10.5T+43T2 1 - 10.5T + 43T^{2}
47 1+47T2 1 + 47T^{2}
53 1+4.24T+53T2 1 + 4.24T + 53T^{2}
59 1−14.2T+59T2 1 - 14.2T + 59T^{2}
61 1+5.68T+61T2 1 + 5.68T + 61T^{2}
67 1+1.12T+67T2 1 + 1.12T + 67T^{2}
71 1+8T+71T2 1 + 8T + 71T^{2}
73 1−0.561T+73T2 1 - 0.561T + 73T^{2}
79 1−2.87T+79T2 1 - 2.87T + 79T^{2}
83 1+17.1T+83T2 1 + 17.1T + 83T^{2}
89 1−10T+89T2 1 - 10T + 89T^{2}
97 1+18.4T+97T2 1 + 18.4T + 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

−8.174758195477873674902437036817, −7.74261461489559210390978058711, −6.94121701775449860723157264232, −5.95372204930711142083955569183, −5.60285197586773223982111565410, −4.56682322034093923531413724618, −4.01190687968821877763013966021, −2.89050009681018490109509802532, −1.88056533104372271822735531205, −0.65933283836549223731925450048, 0.65933283836549223731925450048, 1.88056533104372271822735531205, 2.89050009681018490109509802532, 4.01190687968821877763013966021, 4.56682322034093923531413724618, 5.60285197586773223982111565410, 5.95372204930711142083955569183, 6.94121701775449860723157264232, 7.74261461489559210390978058711, 8.174758195477873674902437036817

Graph of the ZZ-function along the critical line