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Properties

Label 2-336-1.1-c3-0-8
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 + 4·5-s + 7·7-s + 9·9-s + 26·11-s + 2·13-s + 12·15-s − 36·17-s + 76·19-s + 21·21-s + 114·23-s − 109·25-s + 27·27-s + 6·29-s + 256·31-s + 78·33-s + 28·35-s − 86·37-s + 6·39-s + 160·41-s + 220·43-s + 36·45-s − 308·47-s + 49·49-s − 108·51-s + 258·53-s + 104·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.357·5-s + 0.377·7-s + 1/3·9-s + 0.712·11-s + 0.0426·13-s + 0.206·15-s − 0.513·17-s + 0.917·19-s + 0.218·21-s + 1.03·23-s − 0.871·25-s + 0.192·27-s + 0.0384·29-s + 1.48·31-s + 0.411·33-s + 0.135·35-s − 0.382·37-s + 0.0246·39-s + 0.609·41-s + 0.780·43-s + 0.119·45-s − 0.955·47-s + 1/7·49-s − 0.296·51-s + 0.668·53-s + 0.254·55-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 2.7297907862.729790786
L(12)L(\frac12) ≈\approx 2.7297907862.729790786
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 1−4T+p3T2 1 - 4 T + p^{3} T^{2}
11 1−26T+p3T2 1 - 26 T + p^{3} T^{2}
13 1−2T+p3T2 1 - 2 T + p^{3} T^{2}
17 1+36T+p3T2 1 + 36 T + p^{3} T^{2}
19 1−4pT+p3T2 1 - 4 p T + p^{3} T^{2}
23 1−114T+p3T2 1 - 114 T + p^{3} T^{2}
29 1−6T+p3T2 1 - 6 T + p^{3} T^{2}
31 1−256T+p3T2 1 - 256 T + p^{3} T^{2}
37 1+86T+p3T2 1 + 86 T + p^{3} T^{2}
41 1−160T+p3T2 1 - 160 T + p^{3} T^{2}
43 1−220T+p3T2 1 - 220 T + p^{3} T^{2}
47 1+308T+p3T2 1 + 308 T + p^{3} T^{2}
53 1−258T+p3T2 1 - 258 T + p^{3} T^{2}
59 1+264T+p3T2 1 + 264 T + p^{3} T^{2}
61 1−606T+p3T2 1 - 606 T + p^{3} T^{2}
67 1−520T+p3T2 1 - 520 T + p^{3} T^{2}
71 1−286T+p3T2 1 - 286 T + p^{3} T^{2}
73 1+530T+p3T2 1 + 530 T + p^{3} T^{2}
79 1−44T+p3T2 1 - 44 T + p^{3} T^{2}
83 1+1012T+p3T2 1 + 1012 T + p^{3} T^{2}
89 1−768T+p3T2 1 - 768 T + p^{3} T^{2}
97 1−222T+p3T2 1 - 222 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

−11.18202526156211682633030459213, −10.02116976844018075672574879923, −9.256754511391465422248791912843, −8.392643222239171722915915647728, −7.35761502852889108004468996777, −6.34926502809716269383707534680, −5.08227801250171614477862726395, −3.91284713877592745281501384708, −2.59326374861362459413048405266, −1.22145569389880188051329271277, 1.22145569389880188051329271277, 2.59326374861362459413048405266, 3.91284713877592745281501384708, 5.08227801250171614477862726395, 6.34926502809716269383707534680, 7.35761502852889108004468996777, 8.392643222239171722915915647728, 9.256754511391465422248791912843, 10.02116976844018075672574879923, 11.18202526156211682633030459213

Graph of the ZZ-function along the critical line