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Properties

Label 2-336-1.1-c3-0-4
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 + 2·5-s + 7·7-s + 9·9-s + 8·11-s − 42·13-s − 6·15-s − 2·17-s + 124·19-s − 21·21-s − 76·23-s − 121·25-s − 27·27-s + 254·29-s + 72·31-s − 24·33-s + 14·35-s + 398·37-s + 126·39-s + 462·41-s − 212·43-s + 18·45-s + 264·47-s + 49·49-s + 6·51-s − 162·53-s + 16·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.178·5-s + 0.377·7-s + 1/3·9-s + 0.219·11-s − 0.896·13-s − 0.103·15-s − 0.0285·17-s + 1.49·19-s − 0.218·21-s − 0.689·23-s − 0.967·25-s − 0.192·27-s + 1.62·29-s + 0.417·31-s − 0.126·33-s + 0.0676·35-s + 1.76·37-s + 0.517·39-s + 1.75·41-s − 0.751·43-s + 0.0596·45-s + 0.819·47-s + 1/7·49-s + 0.0164·51-s − 0.419·53-s + 0.0392·55-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: 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 1.6171343981.617134398
L(12)L(\frac12) ≈\approx 1.6171343981.617134398
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 1−2T+p3T2 1 - 2 T + p^{3} T^{2}
11 1−8T+p3T2 1 - 8 T + p^{3} T^{2}
13 1+42T+p3T2 1 + 42 T + p^{3} T^{2}
17 1+2T+p3T2 1 + 2 T + p^{3} T^{2}
19 1−124T+p3T2 1 - 124 T + p^{3} T^{2}
23 1+76T+p3T2 1 + 76 T + p^{3} T^{2}
29 1−254T+p3T2 1 - 254 T + p^{3} T^{2}
31 1−72T+p3T2 1 - 72 T + p^{3} T^{2}
37 1−398T+p3T2 1 - 398 T + p^{3} T^{2}
41 1−462T+p3T2 1 - 462 T + p^{3} T^{2}
43 1+212T+p3T2 1 + 212 T + p^{3} T^{2}
47 1−264T+p3T2 1 - 264 T + p^{3} T^{2}
53 1+162T+p3T2 1 + 162 T + p^{3} T^{2}
59 1−772T+p3T2 1 - 772 T + p^{3} T^{2}
61 1−30T+p3T2 1 - 30 T + p^{3} T^{2}
67 1−764T+p3T2 1 - 764 T + p^{3} T^{2}
71 1−236T+p3T2 1 - 236 T + p^{3} T^{2}
73 1−418T+p3T2 1 - 418 T + p^{3} T^{2}
79 1+552T+p3T2 1 + 552 T + p^{3} T^{2}
83 1+1036T+p3T2 1 + 1036 T + p^{3} T^{2}
89 1−30T+p3T2 1 - 30 T + p^{3} T^{2}
97 1+1190T+p3T2 1 + 1190 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

−11.33003161540328336388460586280, −10.06756286809711859051229539139, −9.565871114529556636019698206036, −8.153908778800976919296191311426, −7.30379442001515318330021565113, −6.16188983818076706114829072225, −5.20723213013903529363849410648, −4.17164568001983785458721761358, −2.52299644454531687537926392443, −0.916555713677283105769539900448, 0.916555713677283105769539900448, 2.52299644454531687537926392443, 4.17164568001983785458721761358, 5.20723213013903529363849410648, 6.16188983818076706114829072225, 7.30379442001515318330021565113, 8.153908778800976919296191311426, 9.565871114529556636019698206036, 10.06756286809711859051229539139, 11.33003161540328336388460586280

Graph of the ZZ-function along the critical line