↑

Properties

Label 4-2352e2-1.1-c1e2-0-40
Degree 44
Conductor 55319045531904
Sign 11
Analytic cond. 352.718352.718
Root an. cond. 4.333684.33368
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 + 8·13-s − 4·17-s − 4·19-s + 4·23-s + 5·25-s − 27-s + 4·29-s + 8·31-s + 6·37-s + 8·39-s + 24·41-s − 8·43-s − 8·47-s − 4·51-s − 6·53-s − 4·57-s + 12·59-s − 4·61-s − 4·67-s + 4·69-s + 24·71-s − 8·73-s + 5·75-s − 16·79-s − 81-s + 8·83-s + ⋯
L(s)  = 1  + 0.577·3-s + 2.21·13-s − 0.970·17-s − 0.917·19-s + 0.834·23-s + 25-s − 0.192·27-s + 0.742·29-s + 1.43·31-s + 0.986·37-s + 1.28·39-s + 3.74·41-s − 1.21·43-s − 1.16·47-s − 0.560·51-s − 0.824·53-s − 0.529·57-s + 1.56·59-s − 0.512·61-s − 0.488·67-s + 0.481·69-s + 2.84·71-s − 0.936·73-s + 0.577·75-s − 1.80·79-s − 1/9·81-s + 0.878·83-s + ⋯

Functional equation

Λ(s)=(5531904s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(5531904s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 55319045531904    =    28⋅32⋅742^{8} \cdot 3^{2} \cdot 7^{4}
Sign: 11
Analytic conductor: 352.718352.718
Root analytic conductor: 4.333684.33368
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 5531904, ( :1/2,1/2), 1)(4,\ 5531904,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.1156815474.115681547
L(12)L(\frac12) ≈\approx 4.1156815474.115681547
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
3C2C_2 1−T+T2 1 - T + T^{2}
7 1 1
good5C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
11C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
13C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
17C22C_2^2 1+4T−T2+4pT3+p2T4 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4}
19C22C_2^2 1+4T−3T2+4pT3+p2T4 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4}
23C22C_2^2 1−4T−7T2−4pT3+p2T4 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4}
29C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
31C22C_2^2 1−8T+33T2−8pT3+p2T4 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4}
37C22C_2^2 1−6T−T2−6pT3+p2T4 1 - 6 T - T^{2} - 6 p T^{3} + p^{2} T^{4}
41C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
43C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
47C22C_2^2 1+8T+17T2+8pT3+p2T4 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4}
53C22C_2^2 1+6T−17T2+6pT3+p2T4 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4}
59C22C_2^2 1−12T+85T2−12pT3+p2T4 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4}
61C22C_2^2 1+4T−45T2+4pT3+p2T4 1 + 4 T - 45 T^{2} + 4 p T^{3} + p^{2} T^{4}
67C22C_2^2 1+4T−51T2+4pT3+p2T4 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4}
71C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
73C22C_2^2 1+8T−9T2+8pT3+p2T4 1 + 8 T - 9 T^{2} + 8 p T^{3} + p^{2} T^{4}
79C22C_2^2 1+16T+177T2+16pT3+p2T4 1 + 16 T + 177 T^{2} + 16 p T^{3} + p^{2} T^{4}
83C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
89C22C_2^2 1+4T−73T2+4pT3+p2T4 1 + 4 T - 73 T^{2} + 4 p T^{3} + p^{2} T^{4}
97C2C_2 (1−16T+pT2)2 ( 1 - 16 T + p T^{2} )^{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

−9.007185249115097111108239654797, −8.647414456956895549131623634688, −8.511698711921782716912394959008, −8.264866015736882378246548907461, −7.66567161028868062537591796497, −7.36502253768838549777739257030, −6.72609398785033602049816263354, −6.39996414405837680276046204512, −6.08768281995334148542535537053, −6.00009195851450917541224248759, −5.01931482159563157529158068911, −4.80047134326542298234796424976, −4.26651351612792413368214502230, −3.98012658882641848945314401863, −3.36817634719777948670928236867, −2.97562772599965526826548669934, −2.48651146526484512858525168945, −1.99663048957667265718299553985, −1.10923660635782683757001710235, −0.802156786276615521585753358598, 0.802156786276615521585753358598, 1.10923660635782683757001710235, 1.99663048957667265718299553985, 2.48651146526484512858525168945, 2.97562772599965526826548669934, 3.36817634719777948670928236867, 3.98012658882641848945314401863, 4.26651351612792413368214502230, 4.80047134326542298234796424976, 5.01931482159563157529158068911, 6.00009195851450917541224248759, 6.08768281995334148542535537053, 6.39996414405837680276046204512, 6.72609398785033602049816263354, 7.36502253768838549777739257030, 7.66567161028868062537591796497, 8.264866015736882378246548907461, 8.511698711921782716912394959008, 8.647414456956895549131623634688, 9.007185249115097111108239654797

Graph of the ZZ-function along the critical line