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Properties

Label 4-60e3-1.1-c1e2-0-8
Degree 44
Conductor 216000216000
Sign 11
Analytic cond. 13.772313.7723
Root an. cond. 1.926421.92642
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3-s + 2·4-s + 5-s − 2·6-s + 9-s + 2·10-s − 2·12-s − 15-s − 4·16-s + 2·18-s + 8·19-s + 2·20-s − 4·23-s + 25-s − 27-s + 6·29-s − 2·30-s − 8·32-s + 2·36-s + 16·38-s + 45-s − 8·46-s + 20·47-s + 4·48-s − 2·49-s + 2·50-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 4-s + 0.447·5-s − 0.816·6-s + 1/3·9-s + 0.632·10-s − 0.577·12-s − 0.258·15-s − 16-s + 0.471·18-s + 1.83·19-s + 0.447·20-s − 0.834·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.365·30-s − 1.41·32-s + 1/3·36-s + 2.59·38-s + 0.149·45-s − 1.17·46-s + 2.91·47-s + 0.577·48-s − 2/7·49-s + 0.282·50-s + ⋯

Functional equation

Λ(s)=(216000s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(216000s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 216000216000    =    26⋅33⋅532^{6} \cdot 3^{3} \cdot 5^{3}
Sign: 11
Analytic conductor: 13.772313.7723
Root analytic conductor: 1.926421.92642
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 216000, ( :1/2,1/2), 1)(4,\ 216000,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.2322697053.232269705
L(12)L(\frac12) ≈\approx 3.2322697053.232269705
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−pT+pT2 1 - p T + p T^{2}
3C1C_1 1+T 1 + T
5C1C_1 1−T 1 - T
good7C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
11C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
13C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
17C22C_2^2 1+22T2+p2T4 1 + 22 T^{2} + p^{2} T^{4}
19C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
23C2C_2×\timesC2C_2 (1−2T+pT2)(1+6T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} )
29C2C_2×\timesC2C_2 (1−6T+pT2)(1+pT2) ( 1 - 6 T + p T^{2} )( 1 + p T^{2} )
31C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
37C22C_2^2 1+18T2+p2T4 1 + 18 T^{2} + p^{2} T^{4}
41C22C_2^2 1+70T2+p2T4 1 + 70 T^{2} + p^{2} T^{4}
43C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
47C2C_2×\timesC2C_2 (1−12T+pT2)(1−8T+pT2) ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} )
53C2C_2×\timesC2C_2 (1−6T+pT2)(1−2T+pT2) ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} )
59C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
61C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
67C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
71C2C_2×\timesC2C_2 (1−8T+pT2)(1+pT2) ( 1 - 8 T + p T^{2} )( 1 + p T^{2} )
73C2C_2×\timesC2C_2 (1−4T+pT2)(1+2T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} )
79C22C_2^2 1−82T2+p2T4 1 - 82 T^{2} + p^{2} T^{4}
83C22C_2^2 1−46T2+p2T4 1 - 46 T^{2} + p^{2} T^{4}
89C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
97C2C_2 (1+8T+pT2)(1+18T+pT2) ( 1 + 8 T + p T^{2} )( 1 + 18 T + p T^{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

−9.073530912643278889469956898388, −8.657861741242126463766634676427, −7.88654228264781158140020591353, −7.47252738528849922863843022880, −6.81868368955996367238914359276, −6.58666760429134442590423229824, −5.82490822183852211395449671228, −5.45856204156854412166855237097, −5.35311022889694298747997708124, −4.33995178638783517592705153269, −4.26296913019575463751863880198, −3.37165696296968462837210908940, −2.83061317783178071719510045796, −2.11884394608975336905441282615, −0.977335489126259654610215845871, 0.977335489126259654610215845871, 2.11884394608975336905441282615, 2.83061317783178071719510045796, 3.37165696296968462837210908940, 4.26296913019575463751863880198, 4.33995178638783517592705153269, 5.35311022889694298747997708124, 5.45856204156854412166855237097, 5.82490822183852211395449671228, 6.58666760429134442590423229824, 6.81868368955996367238914359276, 7.47252738528849922863843022880, 7.88654228264781158140020591353, 8.657861741242126463766634676427, 9.073530912643278889469956898388

Graph of the ZZ-function along the critical line