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Properties

Label 4-1110e2-1.1-c1e2-0-6
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 4·5-s − 9-s + 16-s + 4·19-s + 4·20-s + 11·25-s + 16·31-s + 36-s + 12·41-s + 4·45-s + 10·49-s − 12·59-s + 16·61-s − 64-s + 16·71-s − 4·76-s + 32·79-s − 4·80-s + 81-s − 12·89-s − 16·95-s − 11·100-s − 20·101-s + 24·109-s − 22·121-s − 16·124-s + ⋯
L(s)  = 1  − 1/2·4-s − 1.78·5-s − 1/3·9-s + 1/4·16-s + 0.917·19-s + 0.894·20-s + 11/5·25-s + 2.87·31-s + 1/6·36-s + 1.87·41-s + 0.596·45-s + 10/7·49-s − 1.56·59-s + 2.04·61-s − 1/8·64-s + 1.89·71-s − 0.458·76-s + 3.60·79-s − 0.447·80-s + 1/9·81-s − 1.27·89-s − 1.64·95-s − 1.09·100-s − 1.99·101-s + 2.29·109-s − 2·121-s − 1.43·124-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 1.2832390771.283239077
L(12)L(\frac12) ≈\approx 1.2832390771.283239077
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+pT2)2 ( 1 + p T^{2} )^{2}
13C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
17C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
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+pT2)2 ( 1 + p T^{2} )^{2}
31C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
41C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
43C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
47C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
53C22C_2^2 1−102T2+p2T4 1 - 102 T^{2} + p^{2} T^{4}
59C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
61C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
67C22C_2^2 1−118T2+p2T4 1 - 118 T^{2} + p^{2} T^{4}
71C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
73C22C_2^2 1−82T2+p2T4 1 - 82 T^{2} + p^{2} T^{4}
79C2C_2 (1−16T+pT2)2 ( 1 - 16 T + p T^{2} )^{2}
83C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
89C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
97C22C_2^2 1+2T2+p2T4 1 + 2 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

−9.949794568999077430049025979163, −9.687900147503881946912963157314, −9.002733054923657741675495751317, −8.929871208405485281868573329514, −8.168879983998667802800327860234, −8.061045071109727557833480120655, −7.74442124425856362625932106275, −7.29752072306438924333091167403, −6.68001892301503541499825680396, −6.41770390933790760297040453226, −5.79279481821316332130162607663, −5.03142468336652513445714619554, −4.99065158219781470986666377204, −4.24858544144779233391763768170, −3.93008288872750426353207224817, −3.52841878228554168905291947366, −2.77400149945224185587986444726, −2.50563471969673177083762867168, −1.04900108513294705733795950318, −0.64820559882020024916121422129, 0.64820559882020024916121422129, 1.04900108513294705733795950318, 2.50563471969673177083762867168, 2.77400149945224185587986444726, 3.52841878228554168905291947366, 3.93008288872750426353207224817, 4.24858544144779233391763768170, 4.99065158219781470986666377204, 5.03142468336652513445714619554, 5.79279481821316332130162607663, 6.41770390933790760297040453226, 6.68001892301503541499825680396, 7.29752072306438924333091167403, 7.74442124425856362625932106275, 8.061045071109727557833480120655, 8.168879983998667802800327860234, 8.929871208405485281868573329514, 9.002733054923657741675495751317, 9.687900147503881946912963157314, 9.949794568999077430049025979163

Graph of the ZZ-function along the critical line