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Properties

Label 2-336-1.1-c3-0-2
Degree 22
Conductor 336336
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 18·5-s − 7·7-s + 9·9-s + 36·11-s − 34·13-s − 54·15-s + 42·17-s + 124·19-s − 21·21-s + 199·25-s + 27·27-s + 102·29-s + 160·31-s + 108·33-s + 126·35-s + 398·37-s − 102·39-s − 318·41-s + 268·43-s − 162·45-s − 240·47-s + 49·49-s + 126·51-s − 498·53-s − 648·55-s + 372·57-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.60·5-s − 0.377·7-s + 1/3·9-s + 0.986·11-s − 0.725·13-s − 0.929·15-s + 0.599·17-s + 1.49·19-s − 0.218·21-s + 1.59·25-s + 0.192·27-s + 0.653·29-s + 0.926·31-s + 0.569·33-s + 0.608·35-s + 1.76·37-s − 0.418·39-s − 1.21·41-s + 0.950·43-s − 0.536·45-s − 0.744·47-s + 1/7·49-s + 0.345·51-s − 1.29·53-s − 1.58·55-s + 0.864·57-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: 11
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: 00
Selberg data: (2, 336, ( :3/2), 1)(2,\ 336,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.6761428331.676142833
L(12)L(\frac12) ≈\approx 1.6761428331.676142833
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+18T+p3T2 1 + 18 T + p^{3} T^{2}
11 1−36T+p3T2 1 - 36 T + p^{3} T^{2}
13 1+34T+p3T2 1 + 34 T + p^{3} T^{2}
17 1−42T+p3T2 1 - 42 T + p^{3} T^{2}
19 1−124T+p3T2 1 - 124 T + p^{3} T^{2}
23 1+p3T2 1 + p^{3} T^{2}
29 1−102T+p3T2 1 - 102 T + p^{3} T^{2}
31 1−160T+p3T2 1 - 160 T + p^{3} T^{2}
37 1−398T+p3T2 1 - 398 T + p^{3} T^{2}
41 1+318T+p3T2 1 + 318 T + p^{3} T^{2}
43 1−268T+p3T2 1 - 268 T + p^{3} T^{2}
47 1+240T+p3T2 1 + 240 T + p^{3} T^{2}
53 1+498T+p3T2 1 + 498 T + p^{3} T^{2}
59 1−132T+p3T2 1 - 132 T + p^{3} T^{2}
61 1−398T+p3T2 1 - 398 T + p^{3} T^{2}
67 1+92T+p3T2 1 + 92 T + p^{3} T^{2}
71 1−720T+p3T2 1 - 720 T + p^{3} T^{2}
73 1+502T+p3T2 1 + 502 T + p^{3} T^{2}
79 1−1024T+p3T2 1 - 1024 T + p^{3} T^{2}
83 1−204T+p3T2 1 - 204 T + p^{3} T^{2}
89 1−354T+p3T2 1 - 354 T + p^{3} T^{2}
97 1+286T+p3T2 1 + 286 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

−11.43808645846862009442108850231, −10.02564470085939608724930382948, −9.253989239769519311151869418084, −8.108753058559954913654317784987, −7.52899283861443377884794982206, −6.54325801361683782115600262154, −4.84914072840988045227180385771, −3.81155493369603893403057077384, −2.96531272172107702079338172136, −0.877631644606296641782168486277, 0.877631644606296641782168486277, 2.96531272172107702079338172136, 3.81155493369603893403057077384, 4.84914072840988045227180385771, 6.54325801361683782115600262154, 7.52899283861443377884794982206, 8.108753058559954913654317784987, 9.253989239769519311151869418084, 10.02564470085939608724930382948, 11.43808645846862009442108850231

Graph of the ZZ-function along the critical line