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Properties

Label 4-288e2-1.1-c1e2-0-0
Degree 44
Conductor 8294482944
Sign 11
Analytic cond. 5.288585.28858
Root an. cond. 1.516471.51647
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s + 6·25-s − 4·29-s + 16·43-s + 16·47-s + 2·49-s − 4·53-s + 16·67-s − 4·73-s − 4·97-s + 12·101-s + 10·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + 16·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 6·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 1.78·5-s + 6/5·25-s − 0.742·29-s + 2.43·43-s + 2.33·47-s + 2/7·49-s − 0.549·53-s + 1.95·67-s − 0.468·73-s − 0.406·97-s + 1.19·101-s + 0.909·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.32·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.461·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

Λ(s)=(82944s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(82944s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 8294482944    =    210⋅342^{10} \cdot 3^{4}
Sign: 11
Analytic conductor: 5.288585.28858
Root analytic conductor: 1.516471.51647
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 82944, ( :1/2,1/2), 1)(4,\ 82944,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.89362883900.8936288390
L(12)L(\frac12) ≈\approx 0.89362883900.8936288390
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
good5C2C_2×\timesC2C_2 (1+pT2)(1+4T+pT2) ( 1 + p T^{2} )( 1 + 4 T + p T^{2} )
7C2C_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}
13C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
17C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
19C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
23C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
29C2C_2×\timesC2C_2 (1−4T+pT2)(1+8T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} )
31C22C_2^2 1+14T2+p2T4 1 + 14 T^{2} + p^{2} T^{4}
37C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
41C22C_2^2 1−46T2+p2T4 1 - 46 T^{2} + p^{2} T^{4}
43C2C_2×\timesC2C_2 (1−12T+pT2)(1−4T+pT2) ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} )
47C2C_2×\timesC2C_2 (1−12T+pT2)(1−4T+pT2) ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} )
53C2C_2×\timesC2C_2 (1+pT2)(1+4T+pT2) ( 1 + p T^{2} )( 1 + 4 T + p T^{2} )
59C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
61C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
67C2C_2×\timesC2C_2 (1−12T+pT2)(1−4T+pT2) ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} )
71C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
73C2C_2×\timesC2C_2 (1−6T+pT2)(1+10T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} )
79C22C_2^2 1−114T2+p2T4 1 - 114 T^{2} + p^{2} T^{4}
83C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
89C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
97C2C_2 (1+2T+pT2)2 ( 1 + 2 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

−9.565504660524871164429639749353, −9.179229446312042950041884690262, −8.668467112584194245612965425634, −8.093726389751628889142582368081, −7.70886957630587378585693144732, −7.29137810039080651436041493822, −6.90214249458276816357066801553, −6.00310676113253204592181985375, −5.59243936017597682743853641828, −4.77402050493821787769685843993, −4.04928972735847412597755451883, −3.93221712202519314306514798934, −3.07164921560281842898328652629, −2.22083735125283592604328598269, −0.72659832192592192616029365764, 0.72659832192592192616029365764, 2.22083735125283592604328598269, 3.07164921560281842898328652629, 3.93221712202519314306514798934, 4.04928972735847412597755451883, 4.77402050493821787769685843993, 5.59243936017597682743853641828, 6.00310676113253204592181985375, 6.90214249458276816357066801553, 7.29137810039080651436041493822, 7.70886957630587378585693144732, 8.093726389751628889142582368081, 8.668467112584194245612965425634, 9.179229446312042950041884690262, 9.565504660524871164429639749353

Graph of the ZZ-function along the critical line