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Properties

Label 4-1110e2-1.1-c1e2-0-15
Degree 44
Conductor 12321001232100
Sign 11
Analytic cond. 78.559778.5597
Root an. cond. 2.977142.97714
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 4·5-s − 9-s + 8·11-s + 16-s − 4·19-s − 4·20-s + 11·25-s + 16·29-s + 36-s − 4·41-s − 8·44-s − 4·45-s + 10·49-s + 32·55-s − 28·59-s − 24·61-s − 64-s + 4·76-s + 16·79-s + 4·80-s + 81-s + 20·89-s − 16·95-s − 8·99-s − 11·100-s − 12·101-s + ⋯
L(s)  = 1  − 1/2·4-s + 1.78·5-s − 1/3·9-s + 2.41·11-s + 1/4·16-s − 0.917·19-s − 0.894·20-s + 11/5·25-s + 2.97·29-s + 1/6·36-s − 0.624·41-s − 1.20·44-s − 0.596·45-s + 10/7·49-s + 4.31·55-s − 3.64·59-s − 3.07·61-s − 1/8·64-s + 0.458·76-s + 1.80·79-s + 0.447·80-s + 1/9·81-s + 2.11·89-s − 1.64·95-s − 0.804·99-s − 1.09·100-s − 1.19·101-s + ⋯

Functional equation

Λ(s)=(1232100s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1232100s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 12321001232100    =    22⋅32⋅52⋅3722^{2} \cdot 3^{2} \cdot 5^{2} \cdot 37^{2}
Sign: 11
Analytic conductor: 78.559778.5597
Root analytic conductor: 2.977142.97714
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1232100, ( :1/2,1/2), 1)(4,\ 1232100,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.3916784093.391678409
L(12)L(\frac12) ≈\approx 3.3916784093.391678409
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)
bad2C2C_2 1+T2 1 + T^{2}
3C2C_2 1+T2 1 + T^{2}
5C2C_2 1−4T+pT2 1 - 4 T + p T^{2}
37C2C_2 1+T2 1 + T^{2}
good7C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
11C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
13C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
17C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
19C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
23C22C_2^2 1−30T2+p2T4 1 - 30 T^{2} + p^{2} T^{4}
29C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
31C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
41C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
43C22C_2^2 1+58T2+p2T4 1 + 58 T^{2} + p^{2} T^{4}
47C22C_2^2 1−58T2+p2T4 1 - 58 T^{2} + p^{2} T^{4}
53C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
59C2C_2 (1+14T+pT2)2 ( 1 + 14 T + p T^{2} )^{2}
61C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
67C22C_2^2 1−118T2+p2T4 1 - 118 T^{2} + p^{2} T^{4}
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
79C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
83C22C_2^2 1+90T2+p2T4 1 + 90 T^{2} + p^{2} T^{4}
89C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
97C22C_2^2 1−158T2+p2T4 1 - 158 T^{2} + 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

−9.875538461991572270039562668007, −9.575870782330406636055930869150, −9.192659590786573781069738731393, −8.917048079349350748800251563254, −8.712317486699673711262015630400, −8.137299122905775590988903394213, −7.59514694031345826863883307751, −6.89494957778772227203001286030, −6.39288926976370005862739861098, −6.29783850258393869721708026030, −6.13262135834501838138314047609, −5.35507438991081208800447699704, −4.81442077210364497922619153180, −4.46295378969176626891892718463, −4.05192271090513651256314678160, −3.13246285146005863052316712149, −2.92157440145278415652892382268, −1.97868340288436899675780708410, −1.52298635124479814406354454352, −0.904483457506386688431449941298, 0.904483457506386688431449941298, 1.52298635124479814406354454352, 1.97868340288436899675780708410, 2.92157440145278415652892382268, 3.13246285146005863052316712149, 4.05192271090513651256314678160, 4.46295378969176626891892718463, 4.81442077210364497922619153180, 5.35507438991081208800447699704, 6.13262135834501838138314047609, 6.29783850258393869721708026030, 6.39288926976370005862739861098, 6.89494957778772227203001286030, 7.59514694031345826863883307751, 8.137299122905775590988903394213, 8.712317486699673711262015630400, 8.917048079349350748800251563254, 9.192659590786573781069738731393, 9.575870782330406636055930869150, 9.875538461991572270039562668007

Graph of the ZZ-function along the critical line