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Properties

Label 4-78e2-1.1-c1e2-0-2
Degree 44
Conductor 60846084
Sign 11
Analytic cond. 0.3879210.387921
Root an. cond. 0.7891970.789197
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  + 2·3-s − 4-s + 3·9-s − 2·12-s − 6·13-s + 16-s − 4·17-s + 8·23-s + 6·25-s + 4·27-s − 20·29-s − 3·36-s − 12·39-s + 8·43-s + 2·48-s + 10·49-s − 8·51-s + 6·52-s − 12·53-s + 4·61-s − 64-s + 4·68-s + 16·69-s + 12·75-s + 5·81-s − 40·87-s − 8·92-s + ⋯
L(s)  = 1  + 1.15·3-s − 1/2·4-s + 9-s − 0.577·12-s − 1.66·13-s + 1/4·16-s − 0.970·17-s + 1.66·23-s + 6/5·25-s + 0.769·27-s − 3.71·29-s − 1/2·36-s − 1.92·39-s + 1.21·43-s + 0.288·48-s + 10/7·49-s − 1.12·51-s + 0.832·52-s − 1.64·53-s + 0.512·61-s − 1/8·64-s + 0.485·68-s + 1.92·69-s + 1.38·75-s + 5/9·81-s − 4.28·87-s − 0.834·92-s + ⋯

Functional equation

Λ(s)=(6084s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(6084s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 60846084    =    22⋅32⋅1322^{2} \cdot 3^{2} \cdot 13^{2}
Sign: 11
Analytic conductor: 0.3879210.387921
Root analytic conductor: 0.7891970.789197
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 6084, ( :1/2,1/2), 1)(4,\ 6084,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.0692778951.069277895
L(12)L(\frac12) ≈\approx 1.0692778951.069277895
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}
3C1C_1 (1−T)2 ( 1 - T )^{2}
13C2C_2 1+6T+pT2 1 + 6 T + p T^{2}
good5C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
7C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
11C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
17C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
19C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
23C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
29C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
31C22C_2^2 1+38T2+p2T4 1 + 38 T^{2} + p^{2} T^{4}
37C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
41C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
43C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
47C22C_2^2 1+50T2+p2T4 1 + 50 T^{2} + p^{2} T^{4}
53C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
59C22C_2^2 1−102T2+p2T4 1 - 102 T^{2} + p^{2} T^{4}
61C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
67C22C_2^2 1−130T2+p2T4 1 - 130 T^{2} + p^{2} T^{4}
71C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
73C22C_2^2 1−130T2+p2T4 1 - 130 T^{2} + p^{2} T^{4}
79C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
83C22C_2^2 1−150T2+p2T4 1 - 150 T^{2} + p^{2} T^{4}
89C22C_2^2 1−142T2+p2T4 1 - 142 T^{2} + p^{2} T^{4}
97C22C_2^2 1−50T2+p2T4 1 - 50 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

−14.81502284481490964825941282866, −14.21173017908092884179812633889, −13.74217914458887684779145517637, −12.91154339683631396817715304873, −12.89700338987594582710171365238, −12.35987154513940471821027925503, −11.12830875965079958923609098612, −11.10594759747666845413619050874, −10.08531224531131389879503994586, −9.511202414981599377293774012324, −8.984026452504948289969072761145, −8.851745622938911644148999350723, −7.66415074492954259569590107456, −7.41831476835950984450226175229, −6.79786971902108167486082243354, −5.54761891959354108169770988651, −4.85249707345187774716120514420, −4.09590559657322186514860263713, −3.11282312486137849791066275163, −2.16401043466288981441025505835, 2.16401043466288981441025505835, 3.11282312486137849791066275163, 4.09590559657322186514860263713, 4.85249707345187774716120514420, 5.54761891959354108169770988651, 6.79786971902108167486082243354, 7.41831476835950984450226175229, 7.66415074492954259569590107456, 8.851745622938911644148999350723, 8.984026452504948289969072761145, 9.511202414981599377293774012324, 10.08531224531131389879503994586, 11.10594759747666845413619050874, 11.12830875965079958923609098612, 12.35987154513940471821027925503, 12.89700338987594582710171365238, 12.91154339683631396817715304873, 13.74217914458887684779145517637, 14.21173017908092884179812633889, 14.81502284481490964825941282866

Graph of the ZZ-function along the critical line