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Properties

Label 4-1216e2-1.1-c1e2-0-8
Degree 44
Conductor 14786561478656
Sign 11
Analytic cond. 94.280394.2803
Root an. cond. 3.116053.11605
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 2·9-s + 8·13-s − 6·17-s − 7·25-s + 4·29-s − 20·37-s + 20·41-s − 4·45-s − 5·49-s + 8·53-s + 26·61-s + 16·65-s + 18·73-s − 5·81-s − 12·85-s + 24·89-s − 16·97-s + 20·101-s − 20·113-s − 16·117-s + 3·121-s − 26·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 0.894·5-s − 2/3·9-s + 2.21·13-s − 1.45·17-s − 7/5·25-s + 0.742·29-s − 3.28·37-s + 3.12·41-s − 0.596·45-s − 5/7·49-s + 1.09·53-s + 3.32·61-s + 1.98·65-s + 2.10·73-s − 5/9·81-s − 1.30·85-s + 2.54·89-s − 1.62·97-s + 1.99·101-s − 1.88·113-s − 1.47·117-s + 3/11·121-s − 2.32·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

Λ(s)=(1478656s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1478656s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 14786561478656    =    212⋅1922^{12} \cdot 19^{2}
Sign: 11
Analytic conductor: 94.280394.2803
Root analytic conductor: 3.116053.11605
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1478656, ( :1/2,1/2), 1)(4,\ 1478656,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.4153918862.415391886
L(12)L(\frac12) ≈\approx 2.4153918862.415391886
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
19C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
good3C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
5C2C_2 (1−T+pT2)2 ( 1 - T + p T^{2} )^{2}
7C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
11C2C_2 (1−5T+pT2)(1+5T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} )
13C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
17C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
23C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
29C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
31C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
37C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
41C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
43C2C_2 (1−T+pT2)(1+T+pT2) ( 1 - T + p T^{2} )( 1 + T + p T^{2} )
47C2C_2 (1−T+pT2)(1+T+pT2) ( 1 - T + p T^{2} )( 1 + T + p T^{2} )
53C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
59C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
61C2C_2 (1−13T+pT2)2 ( 1 - 13 T + p T^{2} )^{2}
67C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
71C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
73C2C_2 (1−9T+pT2)2 ( 1 - 9 T + p T^{2} )^{2}
79C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
83C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
89C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
97C2C_2 (1+8T+pT2)2 ( 1 + 8 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

−8.121865683855346250917871835791, −7.46802549489449929304458830547, −6.89999375237673531400272497166, −6.39486340421971046323119110374, −6.35171638937504994192666881666, −5.61135260065455155907566257543, −5.55162300458149510222680140561, −4.93085486739421392203871722376, −4.17947794613570531538770424508, −3.73571330395388729025783799541, −3.51647821375355615052320307776, −2.48496537170117614857619865014, −2.22354879090908877160738042469, −1.55214010532178794455093749689, −0.67525660115678139253233977173, 0.67525660115678139253233977173, 1.55214010532178794455093749689, 2.22354879090908877160738042469, 2.48496537170117614857619865014, 3.51647821375355615052320307776, 3.73571330395388729025783799541, 4.17947794613570531538770424508, 4.93085486739421392203871722376, 5.55162300458149510222680140561, 5.61135260065455155907566257543, 6.35171638937504994192666881666, 6.39486340421971046323119110374, 6.89999375237673531400272497166, 7.46802549489449929304458830547, 8.121865683855346250917871835791

Graph of the ZZ-function along the critical line