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Properties

Label 4-1792e2-1.1-c1e2-0-7
Degree 44
Conductor 32112643211264
Sign 11
Analytic cond. 204.752204.752
Root an. cond. 3.782743.78274
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  + 2·7-s + 6·9-s − 12·17-s + 6·25-s + 16·31-s − 4·41-s − 16·47-s + 3·49-s + 12·63-s + 16·71-s − 20·73-s + 32·79-s + 27·81-s + 12·89-s − 12·97-s + 32·103-s + 4·113-s − 24·119-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 72·153-s + 157-s + ⋯
L(s)  = 1  + 0.755·7-s + 2·9-s − 2.91·17-s + 6/5·25-s + 2.87·31-s − 0.624·41-s − 2.33·47-s + 3/7·49-s + 1.51·63-s + 1.89·71-s − 2.34·73-s + 3.60·79-s + 3·81-s + 1.27·89-s − 1.21·97-s + 3.15·103-s + 0.376·113-s − 2.20·119-s + 6/11·121-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 − 5.82·153-s + 0.0798·157-s + ⋯

Functional equation

Λ(s)=(3211264s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3211264s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 32112643211264    =    216⋅722^{16} \cdot 7^{2}
Sign: 11
Analytic conductor: 204.752204.752
Root analytic conductor: 3.782743.78274
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 3211264, ( :1/2,1/2), 1)(4,\ 3211264,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.0593390153.059339015
L(12)L(\frac12) ≈\approx 3.0593390153.059339015
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
7C1C_1 (1−T)2 ( 1 - T )^{2}
good3C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
5C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
11C22C_2^2 1−6T2+p2T4 1 - 6 T^{2} + p^{2} T^{4}
13C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
17C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
19C22C_2^2 1+26T2+p2T4 1 + 26 T^{2} + p^{2} T^{4}
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
31C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
37C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
41C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
43C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
47C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
53C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
59C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
61C22C_2^2 1−86T2+p2T4 1 - 86 T^{2} + p^{2} T^{4}
67C22C_2^2 1−118T2+p2T4 1 - 118 T^{2} + p^{2} T^{4}
71C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
73C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
79C2C_2 (1−16T+pT2)2 ( 1 - 16 T + p T^{2} )^{2}
83C22C_2^2 1−102T2+p2T4 1 - 102 T^{2} + p^{2} T^{4}
89C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
97C2C_2 (1+6T+pT2)2 ( 1 + 6 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

−9.455564899697810592407247663845, −9.105361001425330309830059408909, −8.542412218189265106204449080903, −8.431007374193734365174550384293, −7.919749478658128228399276073598, −7.51503733376692892887126903313, −6.95009324022285998257289448568, −6.74282655436372727131412527977, −6.34838798950487487519238798343, −6.22118238633745572350094155708, −4.98023886211419191598620813075, −4.86400637365457365779841486142, −4.70044314759144490956577086640, −4.24523287725253576647584269256, −3.72728598893114482228370967194, −3.08945340157153625565987591156, −2.30882421588956529155062429466, −2.04685660685479934351914453189, −1.37384435999096037927494630262, −0.69607963127277611925559392862, 0.69607963127277611925559392862, 1.37384435999096037927494630262, 2.04685660685479934351914453189, 2.30882421588956529155062429466, 3.08945340157153625565987591156, 3.72728598893114482228370967194, 4.24523287725253576647584269256, 4.70044314759144490956577086640, 4.86400637365457365779841486142, 4.98023886211419191598620813075, 6.22118238633745572350094155708, 6.34838798950487487519238798343, 6.74282655436372727131412527977, 6.95009324022285998257289448568, 7.51503733376692892887126903313, 7.919749478658128228399276073598, 8.431007374193734365174550384293, 8.542412218189265106204449080903, 9.105361001425330309830059408909, 9.455564899697810592407247663845

Graph of the ZZ-function along the critical line