↑

Properties

Label 4-702e2-1.1-c1e2-0-17
Degree 44
Conductor 492804492804
Sign 11
Analytic cond. 31.421631.4216
Root an. cond. 2.367592.36759
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4·5-s + 2·7-s + 8-s − 4·10-s + 5·11-s + 7·13-s − 2·14-s − 16-s − 6·17-s − 5·22-s − 5·23-s + 2·25-s − 7·26-s + 4·29-s + 20·31-s + 6·34-s + 8·35-s − 6·37-s + 4·40-s + 2·43-s + 5·46-s + 26·47-s + 7·49-s − 2·50-s − 12·53-s + 20·55-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.78·5-s + 0.755·7-s + 0.353·8-s − 1.26·10-s + 1.50·11-s + 1.94·13-s − 0.534·14-s − 1/4·16-s − 1.45·17-s − 1.06·22-s − 1.04·23-s + 2/5·25-s − 1.37·26-s + 0.742·29-s + 3.59·31-s + 1.02·34-s + 1.35·35-s − 0.986·37-s + 0.632·40-s + 0.304·43-s + 0.737·46-s + 3.79·47-s + 49-s − 0.282·50-s − 1.64·53-s + 2.69·55-s + ⋯

Functional equation

Λ(s)=(492804s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 492804 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(492804s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 492804 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 492804492804    =    22⋅36⋅1322^{2} \cdot 3^{6} \cdot 13^{2}
Sign: 11
Analytic conductor: 31.421631.4216
Root analytic conductor: 2.367592.36759
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 492804, ( :1/2,1/2), 1)(4,\ 492804,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.5248971852.524897185
L(12)L(\frac12) ≈\approx 2.5248971852.524897185
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}
3 1 1
13C2C_2 1−7T+pT2 1 - 7 T + p T^{2}
good5C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
7C22C_2^2 1−2T−3T2−2pT3+p2T4 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4}
11C22C_2^2 1−5T+14T2−5pT3+p2T4 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4}
17C22C_2^2 1+6T+19T2+6pT3+p2T4 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4}
19C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
23C22C_2^2 1+5T+2T2+5pT3+p2T4 1 + 5 T + 2 T^{2} + 5 p T^{3} + p^{2} T^{4}
29C22C_2^2 1−4T−13T2−4pT3+p2T4 1 - 4 T - 13 T^{2} - 4 p T^{3} + p^{2} T^{4}
31C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
37C22C_2^2 1+6T−T2+6pT3+p2T4 1 + 6 T - T^{2} + 6 p T^{3} + p^{2} T^{4}
41C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
43C22C_2^2 1−2T−39T2−2pT3+p2T4 1 - 2 T - 39 T^{2} - 2 p T^{3} + p^{2} T^{4}
47C2C_2 (1−13T+pT2)2 ( 1 - 13 T + p T^{2} )^{2}
53C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
59C22C_2^2 1+5T−34T2+5pT3+p2T4 1 + 5 T - 34 T^{2} + 5 p T^{3} + p^{2} T^{4}
61C2C_2 (1−13T+pT2)(1−T+pT2) ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} )
67C22C_2^2 1+10T+33T2+10pT3+p2T4 1 + 10 T + 33 T^{2} + 10 p T^{3} + p^{2} T^{4}
71C22C_2^2 1−8T−7T2−8pT3+p2T4 1 - 8 T - 7 T^{2} - 8 p T^{3} + p^{2} T^{4}
73C2C_2 (1+7T+pT2)2 ( 1 + 7 T + p T^{2} )^{2}
79C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
83C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
89C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
97C22C_2^2 1+17T+192T2+17pT3+p2T4 1 + 17 T + 192 T^{2} + 17 p T^{3} + p^{2} T^{4}
show more
show less
   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

−10.50034096822641886117538386922, −10.31980084018333179725570999432, −9.636891065324781429435712905438, −9.423882734394497125807968345295, −8.806292826762759857454750949693, −8.792763787445136309272789331210, −8.151556563677392041227260128995, −7.972069724836776738087836246446, −6.81627541764353897232085021577, −6.80869108916040031238838285987, −6.05185208606848493716219875607, −6.04861569875053879470978792493, −5.48508711354488382383005589466, −4.58476305409998666295127218709, −4.18538243305469637014629575394, −3.88631947451703367998879168970, −2.66180625516763470328315366953, −2.25387442779167571507744502693, −1.34177446234938717115764028993, −1.21476067643273336426326355457, 1.21476067643273336426326355457, 1.34177446234938717115764028993, 2.25387442779167571507744502693, 2.66180625516763470328315366953, 3.88631947451703367998879168970, 4.18538243305469637014629575394, 4.58476305409998666295127218709, 5.48508711354488382383005589466, 6.04861569875053879470978792493, 6.05185208606848493716219875607, 6.80869108916040031238838285987, 6.81627541764353897232085021577, 7.972069724836776738087836246446, 8.151556563677392041227260128995, 8.792763787445136309272789331210, 8.806292826762759857454750949693, 9.423882734394497125807968345295, 9.636891065324781429435712905438, 10.31980084018333179725570999432, 10.50034096822641886117538386922

Graph of the ZZ-function along the critical line