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Properties

Label 4-1110e2-1.1-c1e2-0-16
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  + 2-s + 3-s + 5-s + 6-s + 5·7-s − 8-s + 10-s + 4·11-s − 5·13-s + 5·14-s + 15-s − 16-s − 2·17-s + 5·19-s + 5·21-s + 4·22-s − 24-s − 5·26-s − 27-s − 18·29-s + 30-s + 8·31-s + 4·33-s − 2·34-s + 5·35-s + 37-s + 5·38-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.447·5-s + 0.408·6-s + 1.88·7-s − 0.353·8-s + 0.316·10-s + 1.20·11-s − 1.38·13-s + 1.33·14-s + 0.258·15-s − 1/4·16-s − 0.485·17-s + 1.14·19-s + 1.09·21-s + 0.852·22-s − 0.204·24-s − 0.980·26-s − 0.192·27-s − 3.34·29-s + 0.182·30-s + 1.43·31-s + 0.696·33-s − 0.342·34-s + 0.845·35-s + 0.164·37-s + 0.811·38-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 5.2513906605.251390660
L(12)L(\frac12) ≈\approx 5.2513906605.251390660
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}
good7C2C_2 (1−4T+pT2)(1−T+pT2) ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} )
11C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
13C2C_2 (1−2T+pT2)(1+7T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 7 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}
19C22C_2^2 1−5T+6T2−5pT3+p2T4 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4}
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C2C_2 (1+9T+pT2)2 ( 1 + 9 T + p T^{2} )^{2}
31C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
41C22C_2^2 1−8T+23T2−8pT3+p2T4 1 - 8 T + 23 T^{2} - 8 p T^{3} + p^{2} T^{4}
43C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
47C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
53C22C_2^2 1−8T+11T2−8pT3+p2T4 1 - 8 T + 11 T^{2} - 8 p T^{3} + p^{2} T^{4}
59C22C_2^2 1−4T−43T2−4pT3+p2T4 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4}
61C2C_2 (1+T+pT2)(1+13T+pT2) ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} )
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+9T+10T2+9pT3+p2T4 1 + 9 T + 10 T^{2} + 9 p T^{3} + p^{2} T^{4}
73C2C_2 (1−16T+pT2)2 ( 1 - 16 T + p T^{2} )^{2}
79C22C_2^2 1−12T+65T2−12pT3+p2T4 1 - 12 T + 65 T^{2} - 12 p T^{3} + p^{2} T^{4}
83C22C_2^2 1−9T−2T2−9pT3+p2T4 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4}
89C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
97C2C_2 (1+4T+pT2)2 ( 1 + 4 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

−9.831343566594205917664549654888, −9.387715447092821216452960063542, −9.350636411152087656773196620287, −8.991774535737113864566293787271, −8.368236106500410927382404112243, −7.78069348500159354035072346839, −7.67836108611901298635361204773, −7.32663558808767185092253249957, −6.67289225518721656527527585044, −6.19557285676697858922175379338, −5.50005982686468226758144248732, −5.40896512500361743712494875341, −4.81426163624664657202230537606, −4.44348654643401444257823562430, −3.91568696558912322202087051379, −3.56268178767474618850704171106, −2.64771732596061523796186109014, −2.22617306963070611371022699356, −1.75047599663205583719936389376, −0.921361052193918108075689182150, 0.921361052193918108075689182150, 1.75047599663205583719936389376, 2.22617306963070611371022699356, 2.64771732596061523796186109014, 3.56268178767474618850704171106, 3.91568696558912322202087051379, 4.44348654643401444257823562430, 4.81426163624664657202230537606, 5.40896512500361743712494875341, 5.50005982686468226758144248732, 6.19557285676697858922175379338, 6.67289225518721656527527585044, 7.32663558808767185092253249957, 7.67836108611901298635361204773, 7.78069348500159354035072346839, 8.368236106500410927382404112243, 8.991774535737113864566293787271, 9.350636411152087656773196620287, 9.387715447092821216452960063542, 9.831343566594205917664549654888

Graph of the ZZ-function along the critical line