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Properties

Label 4-24e4-1.1-c1e2-0-28
Degree 44
Conductor 331776331776
Sign −1-1
Analytic cond. 21.154321.1543
Root an. cond. 2.144612.14461
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  − 4·5-s − 8·13-s + 8·17-s + 8·25-s + 4·29-s − 6·49-s + 12·53-s + 32·65-s − 32·85-s + 8·89-s + 101-s + 103-s + 107-s + 109-s + 113-s − 18·121-s − 20·125-s + 127-s + 131-s + 137-s + 139-s − 16·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 1.78·5-s − 2.21·13-s + 1.94·17-s + 8/5·25-s + 0.742·29-s − 6/7·49-s + 1.64·53-s + 3.96·65-s − 3.47·85-s + 0.847·89-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 1.63·121-s − 1.78·125-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 + ⋯

Functional equation

Λ(s)=(331776s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(331776s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 331776331776    =    212⋅342^{12} \cdot 3^{4}
Sign: −1-1
Analytic conductor: 21.154321.1543
Root analytic conductor: 2.144612.14461
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 331776, ( :1/2,1/2), −1)(4,\ 331776,\ (\ :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
good5C22C_2^2 1+4T+8T2+4pT3+p2T4 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4}
7C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
11C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
13C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
17C22C_2^2 1−8T+32T2−8pT3+p2T4 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4}
19C22C_2^2 1+30T2+p2T4 1 + 30 T^{2} + p^{2} T^{4}
23C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
29C22C_2^2 1−4T+8T2−4pT3+p2T4 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4}
31C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
37C22C_2^2 1−58T2+p2T4 1 - 58 T^{2} + p^{2} T^{4}
41C22C_2^2 1+p2T4 1 + p^{2} T^{4}
43C22C_2^2 1+78T2+p2T4 1 + 78 T^{2} + p^{2} T^{4}
47C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
53C22C_2^2 1−12T+72T2−12pT3+p2T4 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4}
59C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
61C22C_2^2 1−106T2+p2T4 1 - 106 T^{2} + p^{2} T^{4}
67C22C_2^2 1−66T2+p2T4 1 - 66 T^{2} + p^{2} T^{4}
71C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
73C22C_2^2 1−46T2+p2T4 1 - 46 T^{2} + p^{2} T^{4}
79C22C_2^2 1+150T2+p2T4 1 + 150 T^{2} + p^{2} T^{4}
83C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
89C22C_2^2 1−8T+32T2−8pT3+p2T4 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4}
97C22C_2^2 1−158T2+p2T4 1 - 158 T^{2} + p^{2} T^{4}
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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

−13.0203541864, −12.5381507908, −12.2728716993, −12.0227549840, −11.5731744716, −11.4646748129, −10.5503534597, −10.3404197273, −9.93493391536, −9.44709591973, −8.98557544958, −8.29618088815, −7.98383827929, −7.62472305759, −7.30663279167, −6.90167822816, −6.25156000501, −5.45662765031, −5.09031901593, −4.62603189313, −4.02491431772, −3.51276858227, −2.94539460426, −2.33558414449, −1.05860320122, 0, 1.05860320122, 2.33558414449, 2.94539460426, 3.51276858227, 4.02491431772, 4.62603189313, 5.09031901593, 5.45662765031, 6.25156000501, 6.90167822816, 7.30663279167, 7.62472305759, 7.98383827929, 8.29618088815, 8.98557544958, 9.44709591973, 9.93493391536, 10.3404197273, 10.5503534597, 11.4646748129, 11.5731744716, 12.0227549840, 12.2728716993, 12.5381507908, 13.0203541864

Graph of the ZZ-function along the critical line