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Properties

Label 2-4732-1.1-c1-0-52
Degree 22
Conductor 47324732
Sign −1-1
Analytic cond. 37.785237.7852
Root an. cond. 6.146966.14696
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  − 2.42·3-s + 3.63·5-s − 7-s + 2.90·9-s − 3.86·11-s − 8.83·15-s + 1.63·17-s − 2.01·19-s + 2.42·21-s − 2.41·23-s + 8.23·25-s + 0.233·27-s + 4.02·29-s + 9.12·31-s + 9.39·33-s − 3.63·35-s − 12.0·37-s + 9.56·41-s − 9.86·43-s + 10.5·45-s − 8.94·47-s + 49-s − 3.97·51-s − 6.91·53-s − 14.0·55-s + 4.88·57-s + 0.115·59-s + ⋯
L(s)  = 1  − 1.40·3-s + 1.62·5-s − 0.377·7-s + 0.967·9-s − 1.16·11-s − 2.28·15-s + 0.396·17-s − 0.461·19-s + 0.530·21-s − 0.503·23-s + 1.64·25-s + 0.0449·27-s + 0.748·29-s + 1.63·31-s + 1.63·33-s − 0.614·35-s − 1.97·37-s + 1.49·41-s − 1.50·43-s + 1.57·45-s − 1.30·47-s + 0.142·49-s − 0.556·51-s − 0.950·53-s − 1.89·55-s + 0.647·57-s + 0.0150·59-s + ⋯

Functional equation

Λ(s)=(4732s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4732 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(4732s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4732 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 47324732    =    22⋅7⋅1322^{2} \cdot 7 \cdot 13^{2}
Sign: −1-1
Analytic conductor: 37.785237.7852
Root analytic conductor: 6.146966.14696
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 4732, ( :1/2), −1)(2,\ 4732,\ (\ :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
13 1 1
good3 1+2.42T+3T2 1 + 2.42T + 3T^{2}
5 1−3.63T+5T2 1 - 3.63T + 5T^{2}
11 1+3.86T+11T2 1 + 3.86T + 11T^{2}
17 1−1.63T+17T2 1 - 1.63T + 17T^{2}
19 1+2.01T+19T2 1 + 2.01T + 19T^{2}
23 1+2.41T+23T2 1 + 2.41T + 23T^{2}
29 1−4.02T+29T2 1 - 4.02T + 29T^{2}
31 1−9.12T+31T2 1 - 9.12T + 31T^{2}
37 1+12.0T+37T2 1 + 12.0T + 37T^{2}
41 1−9.56T+41T2 1 - 9.56T + 41T^{2}
43 1+9.86T+43T2 1 + 9.86T + 43T^{2}
47 1+8.94T+47T2 1 + 8.94T + 47T^{2}
53 1+6.91T+53T2 1 + 6.91T + 53T^{2}
59 1−0.115T+59T2 1 - 0.115T + 59T^{2}
61 1+4.94T+61T2 1 + 4.94T + 61T^{2}
67 1−9.11T+67T2 1 - 9.11T + 67T^{2}
71 1−0.292T+71T2 1 - 0.292T + 71T^{2}
73 1+1.89T+73T2 1 + 1.89T + 73T^{2}
79 1−5.12T+79T2 1 - 5.12T + 79T^{2}
83 1+11.8T+83T2 1 + 11.8T + 83T^{2}
89 1+2.46T+89T2 1 + 2.46T + 89T^{2}
97 1−17.4T+97T2 1 - 17.4T + 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

−7.920625300912511502909550997279, −6.73882712828888178892444510646, −6.40135606035435276676046584045, −5.74866528573909012576265550915, −5.16539232886272105698109290023, −4.64211836707282827669446897843, −3.16189032925171636683374775300, −2.29961597734273650810113677846, −1.27725833693979325442039651790, 0, 1.27725833693979325442039651790, 2.29961597734273650810113677846, 3.16189032925171636683374775300, 4.64211836707282827669446897843, 5.16539232886272105698109290023, 5.74866528573909012576265550915, 6.40135606035435276676046584045, 6.73882712828888178892444510646, 7.920625300912511502909550997279

Graph of the ZZ-function along the critical line