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Properties

Label 4-7360e2-1.1-c1e2-0-15
Degree 44
Conductor 5416960054169600
Sign 11
Analytic cond. 3453.903453.90
Root an. cond. 7.666157.66615
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 2·5-s + 3·7-s + 4·9-s + 7·11-s − 3·13-s + 6·15-s + 3·17-s − 19-s − 9·21-s − 2·23-s + 3·25-s − 6·27-s − 2·29-s − 5·31-s − 21·33-s − 6·35-s − 16·37-s + 9·39-s − 9·41-s + 4·43-s − 8·45-s + 2·47-s − 4·49-s − 9·51-s + 8·53-s − 14·55-s + ⋯
L(s)  = 1  − 1.73·3-s − 0.894·5-s + 1.13·7-s + 4/3·9-s + 2.11·11-s − 0.832·13-s + 1.54·15-s + 0.727·17-s − 0.229·19-s − 1.96·21-s − 0.417·23-s + 3/5·25-s − 1.15·27-s − 0.371·29-s − 0.898·31-s − 3.65·33-s − 1.01·35-s − 2.63·37-s + 1.44·39-s − 1.40·41-s + 0.609·43-s − 1.19·45-s + 0.291·47-s − 4/7·49-s − 1.26·51-s + 1.09·53-s − 1.88·55-s + ⋯

Functional equation

Λ(s)=(54169600s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 54169600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(54169600s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 54169600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 5416960054169600    =    212⋅52⋅2322^{12} \cdot 5^{2} \cdot 23^{2}
Sign: 11
Analytic conductor: 3453.903453.90
Root analytic conductor: 7.666157.66615
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 54169600, ( :1/2,1/2), 1)(4,\ 54169600,\ (\ :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
5C1C_1 (1+T)2 ( 1 + T )^{2}
23C1C_1 (1+T)2 ( 1 + T )^{2}
good3C4C_4 1+pT+5T2+p2T3+p2T4 1 + p T + 5 T^{2} + p^{2} T^{3} + p^{2} T^{4}
7D4D_{4} 1−3T+13T2−3pT3+p2T4 1 - 3 T + 13 T^{2} - 3 p T^{3} + p^{2} T^{4}
11D4D_{4} 1−7T+31T2−7pT3+p2T4 1 - 7 T + 31 T^{2} - 7 p T^{3} + p^{2} T^{4}
13D4D_{4} 1+3T+25T2+3pT3+p2T4 1 + 3 T + 25 T^{2} + 3 p T^{3} + p^{2} T^{4}
17D4D_{4} 1−3T+7T2−3pT3+p2T4 1 - 3 T + 7 T^{2} - 3 p T^{3} + p^{2} T^{4}
19D4D_{4} 1+T+9T2+pT3+p2T4 1 + T + 9 T^{2} + p T^{3} + p^{2} T^{4}
29D4D_{4} 1+2T+46T2+2pT3+p2T4 1 + 2 T + 46 T^{2} + 2 p T^{3} + p^{2} T^{4}
31D4D_{4} 1+5T+39T2+5pT3+p2T4 1 + 5 T + 39 T^{2} + 5 p T^{3} + p^{2} T^{4}
37C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
41D4D_{4} 1+9T+73T2+9pT3+p2T4 1 + 9 T + 73 T^{2} + 9 p T^{3} + p^{2} T^{4}
43D4D_{4} 1−4T+38T2−4pT3+p2T4 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4}
47D4D_{4} 1−2T+82T2−2pT3+p2T4 1 - 2 T + 82 T^{2} - 2 p T^{3} + p^{2} T^{4}
53D4D_{4} 1−8T+70T2−8pT3+p2T4 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4}
59D4D_{4} 1−14T+154T2−14pT3+p2T4 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4}
61D4D_{4} 1+5T+47T2+5pT3+p2T4 1 + 5 T + 47 T^{2} + 5 p T^{3} + p^{2} T^{4}
67C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
71D4D_{4} 1+29T+349T2+29pT3+p2T4 1 + 29 T + 349 T^{2} + 29 p T^{3} + p^{2} T^{4}
73D4D_{4} 1−10T+54T2−10pT3+p2T4 1 - 10 T + 54 T^{2} - 10 p T^{3} + p^{2} T^{4}
79C22C_2^2 1−50T2+p2T4 1 - 50 T^{2} + p^{2} T^{4}
83D4D_{4} 1+8T+130T2+8pT3+p2T4 1 + 8 T + 130 T^{2} + 8 p T^{3} + p^{2} T^{4}
89C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
97D4D_{4} 1−9T+211T2−9pT3+p2T4 1 - 9 T + 211 T^{2} - 9 p T^{3} + p^{2} T^{4}
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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

−7.50566752367698501639112375208, −7.18584322510690673768686935598, −7.14667009495707518574667036505, −6.72236717090708343800605199924, −6.32509308048312561714142557207, −5.96002464026647663803941422649, −5.48865571271591317563562964026, −5.35342812878046297551027278564, −4.94001319653524944707158482513, −4.68186054890195905046869141343, −4.01259858580927532142217898112, −3.99542928913818942937158401881, −3.55971082638454320585024823335, −3.13634902836226910176117881554, −2.24340247467640058739608131715, −1.84257208463999278687447181318, −1.35479781565662687111880634573, −1.04293745896348535530588738323, 0, 0, 1.04293745896348535530588738323, 1.35479781565662687111880634573, 1.84257208463999278687447181318, 2.24340247467640058739608131715, 3.13634902836226910176117881554, 3.55971082638454320585024823335, 3.99542928913818942937158401881, 4.01259858580927532142217898112, 4.68186054890195905046869141343, 4.94001319653524944707158482513, 5.35342812878046297551027278564, 5.48865571271591317563562964026, 5.96002464026647663803941422649, 6.32509308048312561714142557207, 6.72236717090708343800605199924, 7.14667009495707518574667036505, 7.18584322510690673768686935598, 7.50566752367698501639112375208

Graph of the ZZ-function along the critical line