Properties

Label 4-80e4-1.1-c1e2-0-7
Degree $4$
Conductor $40960000$
Sign $1$
Analytic cond. $2611.64$
Root an. cond. $7.14872$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4·9-s − 4·13-s − 12·17-s + 8·29-s + 4·37-s + 16·41-s + 4·49-s − 4·53-s + 28·61-s − 12·73-s + 7·81-s + 12·89-s − 20·97-s − 12·109-s + 20·113-s + 16·117-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 48·153-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 4/3·9-s − 1.10·13-s − 2.91·17-s + 1.48·29-s + 0.657·37-s + 2.49·41-s + 4/7·49-s − 0.549·53-s + 3.58·61-s − 1.40·73-s + 7/9·81-s + 1.27·89-s − 2.03·97-s − 1.14·109-s + 1.88·113-s + 1.47·117-s + 0.909·121-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 + 3.88·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(40960000\)    =    \(2^{16} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(2611.64\)
Root analytic conductor: \(7.14872\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{6400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 40960000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.765547226\)
\(L(\frac12)\) \(\approx\) \(1.765547226\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 84 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 92 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 116 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 134 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.186467850586397749027986141153, −8.074086687464224393596966877686, −7.28396520912275412880506758914, −7.20712848473981560905399097904, −6.80834289245399036469638759896, −6.46344156646677537755939525367, −6.01947721709480342365195631909, −5.88096670538898794673786482450, −5.22417398028819246446618043501, −5.09611472553510696236255058712, −4.40552100638444449770336114815, −4.33700118840588582943993187673, −4.04779831371830934307694887436, −3.25570041921570547737112506869, −2.80725815646059871585639355420, −2.49571123732536041964462888967, −2.30402609223903457218658652078, −1.77534808113981883384031845367, −0.71298169918710944906666655963, −0.46997589520305112521014395683, 0.46997589520305112521014395683, 0.71298169918710944906666655963, 1.77534808113981883384031845367, 2.30402609223903457218658652078, 2.49571123732536041964462888967, 2.80725815646059871585639355420, 3.25570041921570547737112506869, 4.04779831371830934307694887436, 4.33700118840588582943993187673, 4.40552100638444449770336114815, 5.09611472553510696236255058712, 5.22417398028819246446618043501, 5.88096670538898794673786482450, 6.01947721709480342365195631909, 6.46344156646677537755939525367, 6.80834289245399036469638759896, 7.20712848473981560905399097904, 7.28396520912275412880506758914, 8.074086687464224393596966877686, 8.186467850586397749027986141153

Graph of the $Z$-function along the critical line