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Properties

Label 2-336-1.1-c3-0-11
Degree 22
Conductor 336336
Sign −1-1
Analytic cond. 19.824619.8246
Root an. cond. 4.452484.45248
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 10·5-s + 7·7-s + 9·9-s + 52·11-s − 10·13-s + 30·15-s − 54·17-s + 52·19-s − 21·21-s − 48·23-s − 25·25-s − 27·27-s − 186·29-s − 224·31-s − 156·33-s − 70·35-s + 94·37-s + 30·39-s − 478·41-s + 316·43-s − 90·45-s − 256·47-s + 49·49-s + 162·51-s − 66·53-s − 520·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s + 1.42·11-s − 0.213·13-s + 0.516·15-s − 0.770·17-s + 0.627·19-s − 0.218·21-s − 0.435·23-s − 1/5·25-s − 0.192·27-s − 1.19·29-s − 1.29·31-s − 0.822·33-s − 0.338·35-s + 0.417·37-s + 0.123·39-s − 1.82·41-s + 1.12·43-s − 0.298·45-s − 0.794·47-s + 1/7·49-s + 0.444·51-s − 0.171·53-s − 1.27·55-s + ⋯

Functional equation

Λ(s)=(336s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(336s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 336336    =    24⋅3⋅72^{4} \cdot 3 \cdot 7
Sign: −1-1
Analytic conductor: 19.824619.8246
Root analytic conductor: 4.452484.45248
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 336, ( :3/2), −1)(2,\ 336,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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+pT 1 + p T
7 1−pT 1 - p T
good5 1+2pT+p3T2 1 + 2 p T + p^{3} T^{2}
11 1−52T+p3T2 1 - 52 T + p^{3} T^{2}
13 1+10T+p3T2 1 + 10 T + p^{3} T^{2}
17 1+54T+p3T2 1 + 54 T + p^{3} T^{2}
19 1−52T+p3T2 1 - 52 T + p^{3} T^{2}
23 1+48T+p3T2 1 + 48 T + p^{3} T^{2}
29 1+186T+p3T2 1 + 186 T + p^{3} T^{2}
31 1+224T+p3T2 1 + 224 T + p^{3} T^{2}
37 1−94T+p3T2 1 - 94 T + p^{3} T^{2}
41 1+478T+p3T2 1 + 478 T + p^{3} T^{2}
43 1−316T+p3T2 1 - 316 T + p^{3} T^{2}
47 1+256T+p3T2 1 + 256 T + p^{3} T^{2}
53 1+66T+p3T2 1 + 66 T + p^{3} T^{2}
59 1+420T+p3T2 1 + 420 T + p^{3} T^{2}
61 1−342T+p3T2 1 - 342 T + p^{3} T^{2}
67 1+668T+p3T2 1 + 668 T + p^{3} T^{2}
71 1−272T+p3T2 1 - 272 T + p^{3} T^{2}
73 1+86T+p3T2 1 + 86 T + p^{3} T^{2}
79 1+1360T+p3T2 1 + 1360 T + p^{3} T^{2}
83 1+188T+p3T2 1 + 188 T + p^{3} T^{2}
89 1+366T+p3T2 1 + 366 T + p^{3} T^{2}
97 1−1554T+p3T2 1 - 1554 T + p^{3} T^{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

−11.00438315739502686108727715319, −9.705202424256222519039528277498, −8.818123833378362311540019908528, −7.65784883778203192769729337697, −6.86730016472715386841331681963, −5.72119366176637657418582837601, −4.46893636727600929425998276421, −3.63226233488322435021179897075, −1.62871813279808016586018622682, 0, 1.62871813279808016586018622682, 3.63226233488322435021179897075, 4.46893636727600929425998276421, 5.72119366176637657418582837601, 6.86730016472715386841331681963, 7.65784883778203192769729337697, 8.818123833378362311540019908528, 9.705202424256222519039528277498, 11.00438315739502686108727715319

Graph of the ZZ-function along the critical line