Properties

Label 4-3120e2-1.1-c1e2-0-31
Degree $4$
Conductor $9734400$
Sign $1$
Analytic cond. $620.673$
Root an. cond. $4.99132$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 3·9-s − 4·13-s − 14·17-s − 10·23-s − 25-s + 4·27-s − 20·29-s − 8·39-s + 20·43-s − 11·49-s − 28·51-s + 6·53-s − 6·61-s − 20·69-s − 2·75-s − 22·79-s + 5·81-s − 40·87-s − 20·101-s + 32·103-s − 6·107-s + 12·113-s − 12·117-s + 13·121-s + 127-s + 40·129-s + ⋯
L(s)  = 1  + 1.15·3-s + 9-s − 1.10·13-s − 3.39·17-s − 2.08·23-s − 1/5·25-s + 0.769·27-s − 3.71·29-s − 1.28·39-s + 3.04·43-s − 1.57·49-s − 3.92·51-s + 0.824·53-s − 0.768·61-s − 2.40·69-s − 0.230·75-s − 2.47·79-s + 5/9·81-s − 4.28·87-s − 1.99·101-s + 3.15·103-s − 0.580·107-s + 1.12·113-s − 1.10·117-s + 1.18·121-s + 0.0887·127-s + 3.52·129-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9734400\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(620.673\)
Root analytic conductor: \(4.99132\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 9734400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$ \( ( 1 - T )^{2} \)
5$C_2$ \( 1 + T^{2} \)
13$C_2$ \( 1 + 4 T + p T^{2} \)
good7$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 57 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 21 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 129 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 73 T^{2} + p^{2} T^{4} \)
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

−8.651599867903158058579235104206, −8.069527235610206721637545731093, −7.71543960485720951215901632825, −7.51828269242858243395883582177, −6.97375707545871503590386797504, −6.95191434505101371282007176887, −6.16068857222911978553679690607, −5.90266472969227229826531677338, −5.58381022011482399534427468669, −4.79556756467795226289226496295, −4.46955196212405363697066272667, −4.15743339213029543714365971003, −3.85605536929498218460351649100, −3.37014939302658015710644685594, −2.54107756946499549681844909465, −2.26724809995220455000375157064, −2.09852142466896266384586799509, −1.53461649387954478402058724006, 0, 0, 1.53461649387954478402058724006, 2.09852142466896266384586799509, 2.26724809995220455000375157064, 2.54107756946499549681844909465, 3.37014939302658015710644685594, 3.85605536929498218460351649100, 4.15743339213029543714365971003, 4.46955196212405363697066272667, 4.79556756467795226289226496295, 5.58381022011482399534427468669, 5.90266472969227229826531677338, 6.16068857222911978553679690607, 6.95191434505101371282007176887, 6.97375707545871503590386797504, 7.51828269242858243395883582177, 7.71543960485720951215901632825, 8.069527235610206721637545731093, 8.651599867903158058579235104206

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