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Properties

Label 2-5376-1.1-c1-0-41
Degree 22
Conductor 53765376
Sign −1-1
Analytic cond. 42.927542.9275
Root an. cond. 6.551916.55191
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  − 3-s − 4.10·5-s + 7-s + 9-s − 2.67·11-s − 3.02·13-s + 4.10·15-s − 5.12·17-s + 2.78·19-s − 21-s + 7.12·23-s + 11.8·25-s − 27-s + 8.83·29-s + 1.42·31-s + 2.67·33-s − 4.10·35-s + 1.42·37-s + 3.02·39-s − 5.12·41-s − 2.39·43-s − 4.10·45-s + 9.56·47-s + 49-s + 5.12·51-s − 2.78·53-s + 10.9·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.83·5-s + 0.377·7-s + 0.333·9-s − 0.807·11-s − 0.838·13-s + 1.05·15-s − 1.24·17-s + 0.637·19-s − 0.218·21-s + 1.48·23-s + 2.36·25-s − 0.192·27-s + 1.63·29-s + 0.255·31-s + 0.466·33-s − 0.693·35-s + 0.234·37-s + 0.484·39-s − 0.800·41-s − 0.365·43-s − 0.611·45-s + 1.39·47-s + 0.142·49-s + 0.717·51-s − 0.381·53-s + 1.48·55-s + ⋯

Functional equation

Λ(s)=(5376s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(5376s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 53765376    =    28⋅3⋅72^{8} \cdot 3 \cdot 7
Sign: −1-1
Analytic conductor: 42.927542.9275
Root analytic conductor: 6.551916.55191
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 5376, ( :1/2), −1)(2,\ 5376,\ (\ :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
3 1+T 1 + T
7 1−T 1 - T
good5 1+4.10T+5T2 1 + 4.10T + 5T^{2}
11 1+2.67T+11T2 1 + 2.67T + 11T^{2}
13 1+3.02T+13T2 1 + 3.02T + 13T^{2}
17 1+5.12T+17T2 1 + 5.12T + 17T^{2}
19 1−2.78T+19T2 1 - 2.78T + 19T^{2}
23 1−7.12T+23T2 1 - 7.12T + 23T^{2}
29 1−8.83T+29T2 1 - 8.83T + 29T^{2}
31 1−1.42T+31T2 1 - 1.42T + 31T^{2}
37 1−1.42T+37T2 1 - 1.42T + 37T^{2}
41 1+5.12T+41T2 1 + 5.12T + 41T^{2}
43 1+2.39T+43T2 1 + 2.39T + 43T^{2}
47 1−9.56T+47T2 1 - 9.56T + 47T^{2}
53 1+2.78T+53T2 1 + 2.78T + 53T^{2}
59 1+4T+59T2 1 + 4T + 59T^{2}
61 1−5.17T+61T2 1 - 5.17T + 61T^{2}
67 1+0.244T+67T2 1 + 0.244T + 67T^{2}
71 1+4.27T+71T2 1 + 4.27T + 71T^{2}
73 1−4.15T+73T2 1 - 4.15T + 73T^{2}
79 1+6.25T+79T2 1 + 6.25T + 79T^{2}
83 1+9.35T+83T2 1 + 9.35T + 83T^{2}
89 1+11.2T+89T2 1 + 11.2T + 89T^{2}
97 1−6.69T+97T2 1 - 6.69T + 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

−7.73525556676813909366262227331, −7.13551161482870853532793760031, −6.69046904098119812286592217586, −5.42103868544098599096451669320, −4.67181693767290745726557612868, −4.44931882613903435220505023683, −3.27881814153360258795933919503, −2.56033683328850313102071897212, −0.968094808643267249743844037179, 0, 0.968094808643267249743844037179, 2.56033683328850313102071897212, 3.27881814153360258795933919503, 4.44931882613903435220505023683, 4.67181693767290745726557612868, 5.42103868544098599096451669320, 6.69046904098119812286592217586, 7.13551161482870853532793760031, 7.73525556676813909366262227331

Graph of the ZZ-function along the critical line