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Properties

Label 2-3248-1.1-c1-0-81
Degree 22
Conductor 32483248
Sign −1-1
Analytic cond. 25.935425.9354
Root an. cond. 5.092685.09268
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 + 2.37·5-s + 7-s − 2·9-s − 4.37·11-s − 2.37·13-s + 2.37·15-s + 2·17-s − 7.37·19-s + 21-s − 5.37·23-s + 0.627·25-s − 5·27-s − 29-s + 7.11·31-s − 4.37·33-s + 2.37·35-s − 4·37-s − 2.37·39-s + 0.627·41-s − 8.37·43-s − 4.74·45-s − 9.74·47-s + 49-s + 2·51-s + 9.74·53-s − 10.3·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.06·5-s + 0.377·7-s − 0.666·9-s − 1.31·11-s − 0.657·13-s + 0.612·15-s + 0.485·17-s − 1.69·19-s + 0.218·21-s − 1.12·23-s + 0.125·25-s − 0.962·27-s − 0.185·29-s + 1.27·31-s − 0.761·33-s + 0.400·35-s − 0.657·37-s − 0.379·39-s + 0.0980·41-s − 1.27·43-s − 0.707·45-s − 1.42·47-s + 0.142·49-s + 0.280·51-s + 1.33·53-s − 1.39·55-s + ⋯

Functional equation

Λ(s)=(3248s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(3248s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32483248    =    24⋅7⋅292^{4} \cdot 7 \cdot 29
Sign: −1-1
Analytic conductor: 25.935425.9354
Root analytic conductor: 5.092685.09268
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 3248, ( :1/2), −1)(2,\ 3248,\ (\ :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
29 1+T 1 + T
good3 1−T+3T2 1 - T + 3T^{2}
5 1−2.37T+5T2 1 - 2.37T + 5T^{2}
11 1+4.37T+11T2 1 + 4.37T + 11T^{2}
13 1+2.37T+13T2 1 + 2.37T + 13T^{2}
17 1−2T+17T2 1 - 2T + 17T^{2}
19 1+7.37T+19T2 1 + 7.37T + 19T^{2}
23 1+5.37T+23T2 1 + 5.37T + 23T^{2}
31 1−7.11T+31T2 1 - 7.11T + 31T^{2}
37 1+4T+37T2 1 + 4T + 37T^{2}
41 1−0.627T+41T2 1 - 0.627T + 41T^{2}
43 1+8.37T+43T2 1 + 8.37T + 43T^{2}
47 1+9.74T+47T2 1 + 9.74T + 47T^{2}
53 1−9.74T+53T2 1 - 9.74T + 53T^{2}
59 1+8.74T+59T2 1 + 8.74T + 59T^{2}
61 1+2T+61T2 1 + 2T + 61T^{2}
67 1+7.37T+67T2 1 + 7.37T + 67T^{2}
71 1−5.37T+71T2 1 - 5.37T + 71T^{2}
73 1−11.3T+73T2 1 - 11.3T + 73T^{2}
79 1−15.8T+79T2 1 - 15.8T + 79T^{2}
83 1+10T+83T2 1 + 10T + 83T^{2}
89 1+14.1T+89T2 1 + 14.1T + 89T^{2}
97 1+0.627T+97T2 1 + 0.627T + 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

−8.167602759348633356152768676842, −7.896353527514638694523416005693, −6.69394246064281083795720330422, −5.93769898924469230420339723574, −5.29112941211912007289303150083, −4.50580771573554477325171553715, −3.28491907249612908477283635918, −2.37024763087187552478117523093, −1.92712589790890461493961451724, 0, 1.92712589790890461493961451724, 2.37024763087187552478117523093, 3.28491907249612908477283635918, 4.50580771573554477325171553715, 5.29112941211912007289303150083, 5.93769898924469230420339723574, 6.69394246064281083795720330422, 7.896353527514638694523416005693, 8.167602759348633356152768676842

Graph of the ZZ-function along the critical line