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Properties

Label 4-1095200-1.1-c1e2-0-9
Degree 44
Conductor 10952001095200
Sign 11
Analytic cond. 69.830969.8309
Root an. cond. 2.890752.89075
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 2·5-s + 8-s − 2·9-s + 2·10-s + 4·13-s + 16-s + 12·17-s − 2·18-s + 2·20-s + 3·25-s + 4·26-s + 12·29-s + 32-s + 12·34-s − 2·36-s + 2·37-s + 2·40-s − 12·41-s − 4·45-s − 10·49-s + 3·50-s + 4·52-s + 12·53-s + 12·58-s − 20·61-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s − 2/3·9-s + 0.632·10-s + 1.10·13-s + 1/4·16-s + 2.91·17-s − 0.471·18-s + 0.447·20-s + 3/5·25-s + 0.784·26-s + 2.22·29-s + 0.176·32-s + 2.05·34-s − 1/3·36-s + 0.328·37-s + 0.316·40-s − 1.87·41-s − 0.596·45-s − 1.42·49-s + 0.424·50-s + 0.554·52-s + 1.64·53-s + 1.57·58-s − 2.56·61-s + ⋯

Functional equation

Λ(s)=(1095200s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1095200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1095200s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1095200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 10952001095200    =    25⋅52⋅3722^{5} \cdot 5^{2} \cdot 37^{2}
Sign: 11
Analytic conductor: 69.830969.8309
Root analytic conductor: 2.890752.89075
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1095200, ( :1/2,1/2), 1)(4,\ 1095200,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.9696839884.969683988
L(12)L(\frac12) ≈\approx 4.9696839884.969683988
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 1−T 1 - T
5C1C_1 (1−T)2 ( 1 - T )^{2}
37C1C_1 (1−T)2 ( 1 - T )^{2}
good3C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
7C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
11C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
13C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
17C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
19C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
31C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
41C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
43C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
47C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
53C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
59C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
61C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
67C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
79C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
83C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
89C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
97C2C_2 (1−2T+pT2)2 ( 1 - 2 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

−8.191103303293285704368637540717, −7.38307853599763365662944352653, −7.37364372891229977185623115606, −6.40980423427948262053439128925, −6.24527726803627967890931657857, −5.87742893251240658386511463008, −5.49324095653367495167136231203, −4.80762486474812590647702092154, −4.78046421920466718576600337385, −3.69764756556049564109121180323, −3.29880100176274410685427430205, −3.06956943298945243015211313058, −2.32883355551645860271045039470, −1.44505436450462837187908934432, −1.03875136195910076855341646601, 1.03875136195910076855341646601, 1.44505436450462837187908934432, 2.32883355551645860271045039470, 3.06956943298945243015211313058, 3.29880100176274410685427430205, 3.69764756556049564109121180323, 4.78046421920466718576600337385, 4.80762486474812590647702092154, 5.49324095653367495167136231203, 5.87742893251240658386511463008, 6.24527726803627967890931657857, 6.40980423427948262053439128925, 7.37364372891229977185623115606, 7.38307853599763365662944352653, 8.191103303293285704368637540717

Graph of the ZZ-function along the critical line