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Properties

Label 4-1216e2-1.1-c1e2-0-20
Degree 44
Conductor 14786561478656
Sign −1-1
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 11

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 6·9-s + 8·13-s − 6·17-s − 7·25-s + 16·37-s − 12·45-s − 13·49-s − 4·53-s + 2·61-s + 16·65-s + 10·73-s + 27·81-s − 12·85-s + 12·89-s − 24·97-s − 28·101-s + 4·109-s + 24·113-s − 48·117-s − 13·121-s − 26·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + ⋯
L(s)  = 1  + 0.894·5-s − 2·9-s + 2.21·13-s − 1.45·17-s − 7/5·25-s + 2.63·37-s − 1.78·45-s − 1.85·49-s − 0.549·53-s + 0.256·61-s + 1.98·65-s + 1.17·73-s + 3·81-s − 1.30·85-s + 1.27·89-s − 2.43·97-s − 2.78·101-s + 0.383·109-s + 2.25·113-s − 4.43·117-s − 1.18·121-s − 2.32·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-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: −1-1
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: 11
Selberg data: (4, 1478656, ( :1/2,1/2), −1)(4,\ 1478656,\ (\ :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
19C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
good3C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
5C2C_2 (1−T+pT2)2 ( 1 - T + p T^{2} )^{2}
7C2C_2 (1−T+pT2)(1+T+pT2) ( 1 - T + p T^{2} )( 1 + T + p T^{2} )
11C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 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+pT2)2 ( 1 + p T^{2} )^{2}
31C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
37C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
41C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
43C2C_2 (1−11T+pT2)(1+11T+pT2) ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} )
47C2C_2 (1−7T+pT2)(1+7T+pT2) ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} )
53C2C_2 (1+2T+pT2)2 ( 1 + 2 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−T+pT2)2 ( 1 - T + p T^{2} )^{2}
67C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 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 (1−5T+pT2)2 ( 1 - 5 T + p T^{2} )^{2}
79C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
83C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
89C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
97C2C_2 (1+12T+pT2)2 ( 1 + 12 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

−8.069781997002375745352734843376, −7.22663142743355843577020555382, −6.52327766267142337926450857689, −6.23231971111290599279730913097, −6.07380267050193119311144855802, −5.69887754666040376701153368082, −5.20615537675529298908400867079, −4.57069618429307251486414062571, −4.00622871486103663130279939128, −3.55029150041215641334184982487, −2.95678080105101651849410270529, −2.41972145972916601985658200179, −1.93248528179857739558289949978, −1.10874592215816899307332244272, 0, 1.10874592215816899307332244272, 1.93248528179857739558289949978, 2.41972145972916601985658200179, 2.95678080105101651849410270529, 3.55029150041215641334184982487, 4.00622871486103663130279939128, 4.57069618429307251486414062571, 5.20615537675529298908400867079, 5.69887754666040376701153368082, 6.07380267050193119311144855802, 6.23231971111290599279730913097, 6.52327766267142337926450857689, 7.22663142743355843577020555382, 8.069781997002375745352734843376

Graph of the ZZ-function along the critical line