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Properties

Label 4-1110e2-1.1-c1e2-0-24
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 22

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s − 5-s − 6-s − 3·7-s + 8-s + 10-s − 4·11-s + 5·13-s + 3·14-s − 15-s − 16-s − 2·17-s − 19-s − 3·21-s + 4·22-s − 4·23-s + 24-s − 5·26-s − 27-s − 10·29-s + 30-s − 8·31-s − 4·33-s + 2·34-s + 3·35-s − 37-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 0.447·5-s − 0.408·6-s − 1.13·7-s + 0.353·8-s + 0.316·10-s − 1.20·11-s + 1.38·13-s + 0.801·14-s − 0.258·15-s − 1/4·16-s − 0.485·17-s − 0.229·19-s − 0.654·21-s + 0.852·22-s − 0.834·23-s + 0.204·24-s − 0.980·26-s − 0.192·27-s − 1.85·29-s + 0.182·30-s − 1.43·31-s − 0.696·33-s + 0.342·34-s + 0.507·35-s − 0.164·37-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: 22
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)
bad2C2C_2 1+T+T2 1 + T + T^{2}
3C2C_2 1−T+T2 1 - T + T^{2}
5C2C_2 1+T+T2 1 + T + T^{2}
37C2C_2 1+T+pT2 1 + T + p T^{2}
good7C22C_2^2 1+3T+2T2+3pT3+p2T4 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4}
11C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
13C2C_2 (1−7T+pT2)(1+2T+pT2) ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} )
17C22C_2^2 1+2T−13T2+2pT3+p2T4 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4}
19C2C_2 (1−7T+pT2)(1+8T+pT2) ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} )
23C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
29C2C_2 (1+5T+pT2)2 ( 1 + 5 T + p T^{2} )^{2}
31C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
41C22C_2^2 1+10T+59T2+10pT3+p2T4 1 + 10 T + 59 T^{2} + 10 p T^{3} + p^{2} T^{4}
43C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
47C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
53C22C_2^2 1+6T−17T2+6pT3+p2T4 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4}
59C22C_2^2 1+10T+41T2+10pT3+p2T4 1 + 10 T + 41 T^{2} + 10 p T^{3} + p^{2} T^{4}
61C22C_2^2 1−4T−45T2−4pT3+p2T4 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4}
67C22C_2^2 1+2T−63T2+2pT3+p2T4 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4}
71C22C_2^2 1−T−70T2−pT3+p2T4 1 - T - 70 T^{2} - p T^{3} + p^{2} T^{4}
73C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
79C22C_2^2 1+10T+21T2+10pT3+p2T4 1 + 10 T + 21 T^{2} + 10 p T^{3} + p^{2} T^{4}
83C22C_2^2 1−13T+86T2−13pT3+p2T4 1 - 13 T + 86 T^{2} - 13 p T^{3} + p^{2} T^{4}
89C22C_2^2 1+16T+167T2+16pT3+p2T4 1 + 16 T + 167 T^{2} + 16 p T^{3} + p^{2} T^{4}
97C2C_2 (1−10T+pT2)2 ( 1 - 10 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

−9.469431670114961762918672593103, −9.320053072555493080117409120304, −8.824643971760549332358762440473, −8.277500539204634583840400575198, −8.117795326298462961192202934998, −7.88481517239625029123951184546, −7.12310953446848890489444829599, −6.89180257601775825436961282401, −6.28913424501088577067100030500, −6.01780284818166202160259939497, −5.22761787903985956667123477545, −5.05068949336486866216640329989, −4.20002343178938403091050157208, −3.58224726861927794961352993671, −3.43510292531433566086146437629, −2.95283374525383741297836899869, −1.80379910684588143234779557876, −1.80206367839233783961982228385, 0, 0, 1.80206367839233783961982228385, 1.80379910684588143234779557876, 2.95283374525383741297836899869, 3.43510292531433566086146437629, 3.58224726861927794961352993671, 4.20002343178938403091050157208, 5.05068949336486866216640329989, 5.22761787903985956667123477545, 6.01780284818166202160259939497, 6.28913424501088577067100030500, 6.89180257601775825436961282401, 7.12310953446848890489444829599, 7.88481517239625029123951184546, 8.117795326298462961192202934998, 8.277500539204634583840400575198, 8.824643971760549332358762440473, 9.320053072555493080117409120304, 9.469431670114961762918672593103

Graph of the ZZ-function along the critical line