Properties

Label 4-3920e2-1.1-c1e2-0-0
Degree $4$
Conductor $15366400$
Sign $1$
Analytic cond. $979.774$
Root an. cond. $5.59476$
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  + 3-s − 2·5-s − 9-s + 11-s − 13-s − 2·15-s − 11·17-s − 6·19-s + 2·23-s + 3·25-s + 5·29-s − 4·31-s + 33-s − 39-s − 6·41-s + 6·43-s + 2·45-s + 9·47-s − 11·51-s − 18·53-s − 2·55-s − 6·57-s − 8·59-s + 22·61-s + 2·65-s + 12·67-s + 2·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s − 1/3·9-s + 0.301·11-s − 0.277·13-s − 0.516·15-s − 2.66·17-s − 1.37·19-s + 0.417·23-s + 3/5·25-s + 0.928·29-s − 0.718·31-s + 0.174·33-s − 0.160·39-s − 0.937·41-s + 0.914·43-s + 0.298·45-s + 1.31·47-s − 1.54·51-s − 2.47·53-s − 0.269·55-s − 0.794·57-s − 1.04·59-s + 2.81·61-s + 0.248·65-s + 1.46·67-s + 0.240·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(15366400\)    =    \(2^{8} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(979.774\)
Root analytic conductor: \(5.59476\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3920} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 15366400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6515159017\)
\(L(\frac12)\) \(\approx\) \(0.6515159017\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
7 \( 1 \)
good3$D_{4}$ \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 5 T + 60 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 6 T + 74 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 6 T + 78 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 9 T + 110 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 18 T + 170 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$D_{4}$ \( 1 - 22 T + 226 T^{2} - 22 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$D_{4}$ \( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 11 T + 150 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 + 2 T + 162 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 15 T + 212 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
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

−8.439552931393957149076411583155, −8.413390559583716810039871234536, −8.024683287841735942732738858122, −7.73347908638452966590445425544, −7.04125185857772899732359736917, −6.76024743461157351395031840553, −6.54851986778516274802624188173, −6.43117474919202055708212678422, −5.45371551071751576857553904097, −5.39544520445082610535165483067, −4.68183392952638246326556917763, −4.30969940657618854871906577743, −4.22106000595115389710225185955, −3.71631234761095866487642495601, −3.12164619535452430905706479218, −2.75943905601501150013595501947, −2.12354810181831078238280458730, −2.08676654064415276949547323493, −1.07702418655814939321757275903, −0.23949049112533379500431155917, 0.23949049112533379500431155917, 1.07702418655814939321757275903, 2.08676654064415276949547323493, 2.12354810181831078238280458730, 2.75943905601501150013595501947, 3.12164619535452430905706479218, 3.71631234761095866487642495601, 4.22106000595115389710225185955, 4.30969940657618854871906577743, 4.68183392952638246326556917763, 5.39544520445082610535165483067, 5.45371551071751576857553904097, 6.43117474919202055708212678422, 6.54851986778516274802624188173, 6.76024743461157351395031840553, 7.04125185857772899732359736917, 7.73347908638452966590445425544, 8.024683287841735942732738858122, 8.413390559583716810039871234536, 8.439552931393957149076411583155

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