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Properties

Label 4-1110e2-1.1-c1e2-0-20
Degree 44
Conductor 12321001232100
Sign −1-1
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 11

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 3·9-s + 4·13-s + 16-s − 8·19-s + 25-s − 8·31-s − 3·36-s − 2·37-s + 8·43-s − 14·49-s + 4·52-s + 20·61-s + 64-s − 16·67-s + 20·73-s − 8·76-s − 8·79-s + 9·81-s + 12·97-s + 100-s + 36·109-s − 12·117-s − 6·121-s − 8·124-s + 127-s + 131-s + ⋯
L(s)  = 1  + 1/2·4-s − 9-s + 1.10·13-s + 1/4·16-s − 1.83·19-s + 1/5·25-s − 1.43·31-s − 1/2·36-s − 0.328·37-s + 1.21·43-s − 2·49-s + 0.554·52-s + 2.56·61-s + 1/8·64-s − 1.95·67-s + 2.34·73-s − 0.917·76-s − 0.900·79-s + 81-s + 1.21·97-s + 1/10·100-s + 3.44·109-s − 1.10·117-s − 0.545·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-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: −1-1
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: 11
Selberg data: (4, 1232100, ( :1/2,1/2), −1)(4,\ 1232100,\ (\ :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)
bad2C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
3C2C_2 1+pT2 1 + p T^{2}
5C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
37C1C_1 (1+T)2 ( 1 + T )^{2}
good7C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
11C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
13C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
17C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
19C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
31C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
41C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
43C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
47C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
53C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
59C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
61C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
67C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
79C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
83C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
89C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{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

−7.81059324472633605684417031104, −7.37331018075114938881780514356, −6.87493338522800143592874040603, −6.27790394394954895549345060581, −6.21035074530515276134610094877, −5.72116256637183355409115684013, −5.15564100419176396227847875341, −4.69247680915447547060653501609, −4.00369031444618358374700555311, −3.56437110216151834426597896111, −3.17732882969520207000389302979, −2.26361843784339976106632082372, −2.10151323526117516194563984785, −1.12328850500342707125804122399, 0, 1.12328850500342707125804122399, 2.10151323526117516194563984785, 2.26361843784339976106632082372, 3.17732882969520207000389302979, 3.56437110216151834426597896111, 4.00369031444618358374700555311, 4.69247680915447547060653501609, 5.15564100419176396227847875341, 5.72116256637183355409115684013, 6.21035074530515276134610094877, 6.27790394394954895549345060581, 6.87493338522800143592874040603, 7.37331018075114938881780514356, 7.81059324472633605684417031104

Graph of the ZZ-function along the critical line