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Properties

Label 4-345600-1.1-c1e2-0-2
Degree 44
Conductor 345600345600
Sign 11
Analytic cond. 22.035722.0357
Root an. cond. 2.166612.16661
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  − 3-s − 2·5-s + 9-s + 2·15-s − 25-s − 27-s − 2·45-s + 10·49-s − 20·53-s + 75-s + 16·79-s + 81-s + 8·83-s + 8·107-s + 18·121-s + 12·125-s + 127-s + 131-s + 2·135-s + 137-s + 139-s − 10·147-s + 149-s + 151-s + 157-s + 20·159-s + 163-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s + 1/3·9-s + 0.516·15-s − 1/5·25-s − 0.192·27-s − 0.298·45-s + 10/7·49-s − 2.74·53-s + 0.115·75-s + 1.80·79-s + 1/9·81-s + 0.878·83-s + 0.773·107-s + 1.63·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.172·135-s + 0.0854·137-s + 0.0848·139-s − 0.824·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 1.58·159-s + 0.0783·163-s + ⋯

Functional equation

Λ(s)=(345600s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 345600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(345600s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 345600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 345600345600    =    29⋅33⋅522^{9} \cdot 3^{3} \cdot 5^{2}
Sign: 11
Analytic conductor: 22.035722.0357
Root analytic conductor: 2.166612.16661
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 345600, ( :1/2,1/2), 1)(4,\ 345600,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.91760641800.9176064180
L(12)L(\frac12) ≈\approx 0.91760641800.9176064180
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
3C1C_1 1+T 1 + T
5C2C_2 1+2T+pT2 1 + 2 T + p T^{2}
good7C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
11C22C_2^2 1−18T2+p2T4 1 - 18 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 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
19C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
23C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
29C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
31C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
37C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
41C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
43C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
47C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
53C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
59C22C_2^2 1−82T2+p2T4 1 - 82 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} )
67C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
71C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
73C22C_2^2 1−130T2+p2T4 1 - 130 T^{2} + p^{2} T^{4}
79C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
83C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
89C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
97C2C_2 (1−18T+pT2)(1+18T+pT2) ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{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

−8.720556122204919940787114226817, −8.127280331995193708455482208382, −7.85101984287289917616469398437, −7.34999160539775999043577503268, −6.95203567399306177371933355135, −6.31795351810838792684187487407, −5.99939825891420261041909475864, −5.37595707552247311413898151676, −4.76079991274913578801014190589, −4.46360011835707575831995568528, −3.70428071362386722458673170717, −3.37528171703364868655624815248, −2.48435940156910725970454962683, −1.65720388043715485575795658732, −0.57100453902310593927136460717, 0.57100453902310593927136460717, 1.65720388043715485575795658732, 2.48435940156910725970454962683, 3.37528171703364868655624815248, 3.70428071362386722458673170717, 4.46360011835707575831995568528, 4.76079991274913578801014190589, 5.37595707552247311413898151676, 5.99939825891420261041909475864, 6.31795351810838792684187487407, 6.95203567399306177371933355135, 7.34999160539775999043577503268, 7.85101984287289917616469398437, 8.127280331995193708455482208382, 8.720556122204919940787114226817

Graph of the ZZ-function along the critical line