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Properties

Label 4-360e2-1.1-c1e2-0-37
Degree 44
Conductor 129600129600
Sign −1-1
Analytic cond. 8.263408.26340
Root an. cond. 1.695461.69546
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·13-s + 25-s − 8·37-s + 2·49-s − 20·61-s + 4·73-s − 20·97-s − 20·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 2.21·13-s + 1/5·25-s − 1.31·37-s + 2/7·49-s − 2.56·61-s + 0.468·73-s − 2.03·97-s − 1.91·109-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯

Functional equation

Λ(s)=(129600s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(129600s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 129600129600    =    26⋅34⋅522^{6} \cdot 3^{4} \cdot 5^{2}
Sign: −1-1
Analytic conductor: 8.263408.26340
Root analytic conductor: 1.695461.69546
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 129600, ( :1/2,1/2), −1)(4,\ 129600,\ (\ :1/2, 1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2 1 1
3 1 1
5C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
good7C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
11C22C_2^2 1+10T2+p2T4 1 + 10 T^{2} + p^{2} T^{4}
13C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
17C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
19C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
23C22C_2^2 1+34T2+p2T4 1 + 34 T^{2} + p^{2} T^{4}
29C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
31C22C_2^2 1−50T2+p2T4 1 - 50 T^{2} + p^{2} T^{4}
37C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
41C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
43C22C_2^2 1−38T2+p2T4 1 - 38 T^{2} + p^{2} T^{4}
47C22C_2^2 1+82T2+p2T4 1 + 82 T^{2} + p^{2} T^{4}
53C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
59C22C_2^2 1+106T2+p2T4 1 + 106 T^{2} + p^{2} T^{4}
61C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
67C22C_2^2 1−86T2+p2T4 1 - 86 T^{2} + p^{2} T^{4}
71C22C_2^2 1−50T2+p2T4 1 - 50 T^{2} + p^{2} T^{4}
73C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
79C22C_2^2 1−50T2+p2T4 1 - 50 T^{2} + p^{2} T^{4}
83C22C_2^2 1+58T2+p2T4 1 + 58 T^{2} + p^{2} T^{4}
89C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
97C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
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   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−9.206150334203433277032805518075, −8.726729716648949012339085952986, −8.115025891174599029868796914500, −7.56955194003007377787059704536, −7.27811208276789072767353456205, −6.72982410874228824953087586964, −6.20472733169143353996301240057, −5.38965699084713833393947846363, −5.08672178038356679981437010710, −4.52031162325514830676751253759, −3.87096965336205369146462548095, −2.98620008521077513444486906212, −2.50155778785474847198298296681, −1.59933090642997300853612755639, 0, 1.59933090642997300853612755639, 2.50155778785474847198298296681, 2.98620008521077513444486906212, 3.87096965336205369146462548095, 4.52031162325514830676751253759, 5.08672178038356679981437010710, 5.38965699084713833393947846363, 6.20472733169143353996301240057, 6.72982410874228824953087586964, 7.27811208276789072767353456205, 7.56955194003007377787059704536, 8.115025891174599029868796914500, 8.726729716648949012339085952986, 9.206150334203433277032805518075

Graph of the ZZ-function along the critical line