↑

Properties

Label 4-1216e2-1.1-c1e2-0-1
Degree 44
Conductor 14786561478656
Sign 11
Analytic cond. 94.280394.2803
Root an. cond. 3.116053.11605
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  + 3-s − 5-s + 3·9-s − 8·11-s − 13-s − 15-s − 3·17-s − 8·19-s + 5·23-s + 5·25-s + 8·27-s + 7·29-s − 8·31-s − 8·33-s − 20·37-s − 39-s + 5·41-s + 5·43-s − 3·45-s − 7·47-s − 14·49-s − 3·51-s + 11·53-s + 8·55-s − 8·57-s − 3·59-s + 11·61-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 9-s − 2.41·11-s − 0.277·13-s − 0.258·15-s − 0.727·17-s − 1.83·19-s + 1.04·23-s + 25-s + 1.53·27-s + 1.29·29-s − 1.43·31-s − 1.39·33-s − 3.28·37-s − 0.160·39-s + 0.780·41-s + 0.762·43-s − 0.447·45-s − 1.02·47-s − 2·49-s − 0.420·51-s + 1.51·53-s + 1.07·55-s − 1.05·57-s − 0.390·59-s + 1.40·61-s + ⋯

Functional equation

Λ(s)=(1478656s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1478656s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 14786561478656    =    212⋅1922^{12} \cdot 19^{2}
Sign: 11
Analytic conductor: 94.280394.2803
Root analytic conductor: 3.116053.11605
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1478656, ( :1/2,1/2), 1)(4,\ 1478656,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.1650919051.165091905
L(12)L(\frac12) ≈\approx 1.1650919051.165091905
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)
bad2 1 1
19C2C_2 1+8T+pT2 1 + 8 T + p T^{2}
good3C22C_2^2 1−T−2T2−pT3+p2T4 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4}
5C22C_2^2 1+T−4T2+pT3+p2T4 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4}
7C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
11C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
13C22C_2^2 1+T−12T2+pT3+p2T4 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4}
17C22C_2^2 1+3T−8T2+3pT3+p2T4 1 + 3 T - 8 T^{2} + 3 p T^{3} + 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 1−7T+20T2−7pT3+p2T4 1 - 7 T + 20 T^{2} - 7 p T^{3} + p^{2} T^{4}
31C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
37C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
41C22C_2^2 1−5T−16T2−5pT3+p2T4 1 - 5 T - 16 T^{2} - 5 p T^{3} + p^{2} T^{4}
43C2C_2 (1−13T+pT2)(1+8T+pT2) ( 1 - 13 T + p T^{2} )( 1 + 8 T + p T^{2} )
47C22C_2^2 1+7T+2T2+7pT3+p2T4 1 + 7 T + 2 T^{2} + 7 p T^{3} + p^{2} T^{4}
53C22C_2^2 1−11T+68T2−11pT3+p2T4 1 - 11 T + 68 T^{2} - 11 p T^{3} + p^{2} T^{4}
59C22C_2^2 1+3T−50T2+3pT3+p2T4 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4}
61C22C_2^2 1−11T+60T2−11pT3+p2T4 1 - 11 T + 60 T^{2} - 11 p T^{3} + p^{2} T^{4}
67C22C_2^2 1−3T−58T2−3pT3+p2T4 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4}
71C22C_2^2 1−11T+50T2−11pT3+p2T4 1 - 11 T + 50 T^{2} - 11 p T^{3} + p^{2} T^{4}
73C22C_2^2 1+15T+152T2+15pT3+p2T4 1 + 15 T + 152 T^{2} + 15 p T^{3} + p^{2} T^{4}
79C2C_2 (1−4T+pT2)(1+17T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 17 T + p T^{2} )
83C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
89C22C_2^2 1+3T−80T2+3pT3+p2T4 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4}
97C2C_2 (1−19T+pT2)(1+14T+pT2) ( 1 - 19 T + p T^{2} )( 1 + 14 T + p T^{2} )
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

−10.30635681697216334422864240023, −9.513383454135585988270874709104, −8.857571436506208770288316013988, −8.690712882522587051949739378897, −8.325139298921436122740996078547, −8.108139705660945225610688135763, −7.30575442479180154324138690813, −7.04099428933538834886565069078, −7.00387469099120421651974817908, −6.26606043798695773688729472289, −5.67637069027213819610780671363, −4.93970543655673286845490453258, −4.86486881921466456229017161779, −4.53854471935038728507819800624, −3.69116018372694259187518944766, −3.28546026377520809595273889941, −2.66361088398248151877003270588, −2.29035348611369716291379545667, −1.64565947226456425102207556606, −0.42643685515750211111448020377, 0.42643685515750211111448020377, 1.64565947226456425102207556606, 2.29035348611369716291379545667, 2.66361088398248151877003270588, 3.28546026377520809595273889941, 3.69116018372694259187518944766, 4.53854471935038728507819800624, 4.86486881921466456229017161779, 4.93970543655673286845490453258, 5.67637069027213819610780671363, 6.26606043798695773688729472289, 7.00387469099120421651974817908, 7.04099428933538834886565069078, 7.30575442479180154324138690813, 8.108139705660945225610688135763, 8.325139298921436122740996078547, 8.690712882522587051949739378897, 8.857571436506208770288316013988, 9.513383454135585988270874709104, 10.30635681697216334422864240023

Graph of the ZZ-function along the critical line