↑

Properties

Label 4-1216e2-1.1-c1e2-0-22
Degree 44
Conductor 14786561478656
Sign −1-1
Analytic cond. 94.280394.2803
Root an. cond. 3.116053.11605
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 9-s + 4·13-s − 6·17-s − 6·25-s + 12·29-s + 8·37-s − 5·49-s − 12·53-s − 8·61-s − 2·73-s − 8·81-s − 12·89-s − 12·97-s − 12·101-s − 12·109-s + 12·113-s + 4·117-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 6·153-s + 157-s + 163-s + ⋯
L(s)  = 1  + 1/3·9-s + 1.10·13-s − 1.45·17-s − 6/5·25-s + 2.22·29-s + 1.31·37-s − 5/7·49-s − 1.64·53-s − 1.02·61-s − 0.234·73-s − 8/9·81-s − 1.27·89-s − 1.21·97-s − 1.19·101-s − 1.14·109-s + 1.12·113-s + 0.369·117-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 0.485·153-s + 0.0798·157-s + 0.0783·163-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: −1-1
Analytic conductor: 94.280394.2803
Root analytic conductor: 3.116053.11605
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 1478656, ( :1/2,1/2), −1)(4,\ 1478656,\ (\ :1/2, 1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+T2 1 + T^{2}
good3C22C_2^2 1−T2+p2T4 1 - T^{2} + p^{2} T^{4}
5C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
7C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
11C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
13C2C_2×\timesC2C_2 (1−5T+pT2)(1+T+pT2) ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} )
17C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
23C22C_2^2 1−27T2+p2T4 1 - 27 T^{2} + p^{2} T^{4}
29C2C_2×\timesC2C_2 (1−9T+pT2)(1−3T+pT2) ( 1 - 9 T + p T^{2} )( 1 - 3 T + p T^{2} )
31C22C_2^2 1+14T2+p2T4 1 + 14 T^{2} + p^{2} T^{4}
37C2C_2×\timesC2C_2 (1−10T+pT2)(1+2T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} )
41C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
43C22C_2^2 1+62T2+p2T4 1 + 62 T^{2} + p^{2} T^{4}
47C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
53C2C_2×\timesC2C_2 (1−T+pT2)(1+13T+pT2) ( 1 - T + p T^{2} )( 1 + 13 T + p T^{2} )
59C22C_2^2 1−57T2+p2T4 1 - 57 T^{2} + p^{2} T^{4}
61C2C_2×\timesC2C_2 (1−2T+pT2)(1+10T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} )
67C22C_2^2 1−97T2+p2T4 1 - 97 T^{2} + p^{2} T^{4}
71C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
73C2C_2×\timesC2C_2 (1−11T+pT2)(1+13T+pT2) ( 1 - 11 T + p T^{2} )( 1 + 13 T + p T^{2} )
79C22C_2^2 1+110T2+p2T4 1 + 110 T^{2} + p^{2} T^{4}
83C22C_2^2 1+58T2+p2T4 1 + 58 T^{2} + p^{2} T^{4}
89C2C_2×\timesC2C_2 (1−2T+pT2)(1+14T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} )
97C2C_2×\timesC2C_2 (1+pT2)(1+12T+pT2) ( 1 + p T^{2} )( 1 + 12 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

−7.79479479553236507364486493732, −7.27774099213358132015809486803, −6.68957202721806193978273217581, −6.35014987329731172636910476357, −6.18099911021454210276504007249, −5.58353039983297327223378157199, −4.90711614985539699610999865212, −4.52831809305135001637386747581, −4.13546939951366041525236225079, −3.68248419208624114766605593517, −2.86217733411437857444907232853, −2.60478905040558338261948072875, −1.68216410306367556145215742738, −1.19792738995265454495203065562, 0, 1.19792738995265454495203065562, 1.68216410306367556145215742738, 2.60478905040558338261948072875, 2.86217733411437857444907232853, 3.68248419208624114766605593517, 4.13546939951366041525236225079, 4.52831809305135001637386747581, 4.90711614985539699610999865212, 5.58353039983297327223378157199, 6.18099911021454210276504007249, 6.35014987329731172636910476357, 6.68957202721806193978273217581, 7.27774099213358132015809486803, 7.79479479553236507364486493732

Graph of the ZZ-function along the critical line