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Properties

Label 4-640e2-1.1-c1e2-0-9
Degree 44
Conductor 409600409600
Sign 11
Analytic cond. 26.116426.1164
Root an. cond. 2.260622.26062
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  + 6·9-s − 4·17-s + 25-s − 12·41-s + 2·49-s + 12·73-s + 27·81-s + 12·89-s + 28·97-s + 36·113-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 24·153-s + 157-s + 163-s + 167-s − 22·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 2·9-s − 0.970·17-s + 1/5·25-s − 1.87·41-s + 2/7·49-s + 1.40·73-s + 3·81-s + 1.27·89-s + 2.84·97-s + 3.38·113-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 − 1.94·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

Functional equation

Λ(s)=(409600s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 409600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(409600s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 409600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 409600409600    =    214⋅522^{14} \cdot 5^{2}
Sign: 11
Analytic conductor: 26.116426.1164
Root analytic conductor: 2.260622.26062
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 409600, ( :1/2,1/2), 1)(4,\ 409600,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.1191205212.119120521
L(12)L(\frac12) ≈\approx 2.1191205212.119120521
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
5C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
good3C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
7C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
11C22C_2^2 1−6T2+p2T4 1 - 6 T^{2} + p^{2} T^{4}
13C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
17C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
19C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
23C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
29C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
31C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
37C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
41C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
43C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
47C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
53C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
59C22C_2^2 1−102T2+p2T4 1 - 102 T^{2} + p^{2} T^{4}
61C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
67C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
79C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
83C22C_2^2 1+90T2+p2T4 1 + 90 T^{2} + p^{2} T^{4}
89C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
97C2C_2 (1−14T+pT2)2 ( 1 - 14 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

−8.747632885960857457554511369924, −8.135119204590986840147380543289, −7.63681743416551529797305844505, −7.22245389961459731580036601376, −6.82203881238171856760208727134, −6.46388932185968270212586100536, −5.93658508702551377717138893647, −5.11545329537655183854292146626, −4.72991726389300358426403430935, −4.39259698449799546170391543910, −3.66846502775993424438644585598, −3.30408489711525981702747105766, −2.16341080804733184851514212185, −1.83998957870676792591407425914, −0.833094286916894356940394732605, 0.833094286916894356940394732605, 1.83998957870676792591407425914, 2.16341080804733184851514212185, 3.30408489711525981702747105766, 3.66846502775993424438644585598, 4.39259698449799546170391543910, 4.72991726389300358426403430935, 5.11545329537655183854292146626, 5.93658508702551377717138893647, 6.46388932185968270212586100536, 6.82203881238171856760208727134, 7.22245389961459731580036601376, 7.63681743416551529797305844505, 8.135119204590986840147380543289, 8.747632885960857457554511369924

Graph of the ZZ-function along the critical line