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Properties

Label 2-6336-1.1-c1-0-52
Degree 22
Conductor 63366336
Sign 11
Analytic cond. 50.593250.5932
Root an. cond. 7.112897.11289
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  + 3.56·5-s + 3.12·7-s − 11-s + 5.12·13-s − 2·17-s + 4·19-s − 2.43·23-s + 7.68·25-s − 5.12·29-s − 5.56·31-s + 11.1·35-s + 7.56·37-s + 1.12·41-s + 7.12·43-s − 8·47-s + 2.75·49-s + 12.2·53-s − 3.56·55-s + 7.80·59-s − 1.12·61-s + 18.2·65-s − 9.56·67-s + 8.68·71-s + 5.12·73-s − 3.12·77-s − 11.1·79-s + 0.876·83-s + ⋯
L(s)  = 1  + 1.59·5-s + 1.18·7-s − 0.301·11-s + 1.42·13-s − 0.485·17-s + 0.917·19-s − 0.508·23-s + 1.53·25-s − 0.951·29-s − 0.998·31-s + 1.88·35-s + 1.24·37-s + 0.175·41-s + 1.08·43-s − 1.16·47-s + 0.393·49-s + 1.68·53-s − 0.480·55-s + 1.01·59-s − 0.143·61-s + 2.26·65-s − 1.16·67-s + 1.03·71-s + 0.599·73-s − 0.355·77-s − 1.25·79-s + 0.0962·83-s + ⋯

Functional equation

Λ(s)=(6336s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(6336s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 63366336    =    26⋅32⋅112^{6} \cdot 3^{2} \cdot 11
Sign: 11
Analytic conductor: 50.593250.5932
Root analytic conductor: 7.112897.11289
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 6336, ( :1/2), 1)(2,\ 6336,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.6242031283.624203128
L(12)L(\frac12) ≈\approx 3.6242031283.624203128
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 1
11 1+T 1 + T
good5 1−3.56T+5T2 1 - 3.56T + 5T^{2}
7 1−3.12T+7T2 1 - 3.12T + 7T^{2}
13 1−5.12T+13T2 1 - 5.12T + 13T^{2}
17 1+2T+17T2 1 + 2T + 17T^{2}
19 1−4T+19T2 1 - 4T + 19T^{2}
23 1+2.43T+23T2 1 + 2.43T + 23T^{2}
29 1+5.12T+29T2 1 + 5.12T + 29T^{2}
31 1+5.56T+31T2 1 + 5.56T + 31T^{2}
37 1−7.56T+37T2 1 - 7.56T + 37T^{2}
41 1−1.12T+41T2 1 - 1.12T + 41T^{2}
43 1−7.12T+43T2 1 - 7.12T + 43T^{2}
47 1+8T+47T2 1 + 8T + 47T^{2}
53 1−12.2T+53T2 1 - 12.2T + 53T^{2}
59 1−7.80T+59T2 1 - 7.80T + 59T^{2}
61 1+1.12T+61T2 1 + 1.12T + 61T^{2}
67 1+9.56T+67T2 1 + 9.56T + 67T^{2}
71 1−8.68T+71T2 1 - 8.68T + 71T^{2}
73 1−5.12T+73T2 1 - 5.12T + 73T^{2}
79 1+11.1T+79T2 1 + 11.1T + 79T^{2}
83 1−0.876T+83T2 1 - 0.876T + 83T^{2}
89 1+2.68T+89T2 1 + 2.68T + 89T^{2}
97 1−15.5T+97T2 1 - 15.5T + 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

−8.081498127125947361477778328167, −7.36813072302160515271592336962, −6.47862077965142934251158356936, −5.67840920998498083749479687348, −5.49306341965071616054522685808, −4.51475117725359205918001901078, −3.64417200868825107703509524537, −2.49733713128358253641998926041, −1.80791720950501953299311220262, −1.09393554670933154497075004180, 1.09393554670933154497075004180, 1.80791720950501953299311220262, 2.49733713128358253641998926041, 3.64417200868825107703509524537, 4.51475117725359205918001901078, 5.49306341965071616054522685808, 5.67840920998498083749479687348, 6.47862077965142934251158356936, 7.36813072302160515271592336962, 8.081498127125947361477778328167

Graph of the ZZ-function along the critical line