Properties

Label 4-5290e2-1.1-c1e2-0-9
Degree $4$
Conductor $27984100$
Sign $1$
Analytic cond. $1784.29$
Root an. cond. $6.49929$
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  − 2·2-s − 3-s + 3·4-s + 2·5-s + 2·6-s − 7-s − 4·8-s − 4·10-s − 3·11-s − 3·12-s + 7·13-s + 2·14-s − 2·15-s + 5·16-s + 3·17-s − 7·19-s + 6·20-s + 21-s + 6·22-s + 4·24-s + 3·25-s − 14·26-s − 2·27-s − 3·28-s + 6·29-s + 4·30-s + 7·31-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 3/2·4-s + 0.894·5-s + 0.816·6-s − 0.377·7-s − 1.41·8-s − 1.26·10-s − 0.904·11-s − 0.866·12-s + 1.94·13-s + 0.534·14-s − 0.516·15-s + 5/4·16-s + 0.727·17-s − 1.60·19-s + 1.34·20-s + 0.218·21-s + 1.27·22-s + 0.816·24-s + 3/5·25-s − 2.74·26-s − 0.384·27-s − 0.566·28-s + 1.11·29-s + 0.730·30-s + 1.25·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27984100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27984100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(27984100\)    =    \(2^{2} \cdot 5^{2} \cdot 23^{4}\)
Sign: $1$
Analytic conductor: \(1784.29\)
Root analytic conductor: \(6.49929\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{5290} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 27984100,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
23 \( 1 \)
good3$D_{4}$ \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + T + 9 T^{2} + p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 3 T + 19 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 7 T + 33 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 3 T + 31 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 7 T + 45 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 6 T + 46 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 7 T + 27 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 + 9 T + 97 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 18 T + 154 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 + 18 T + 178 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 7 T + 87 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 3 T + 97 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 2 T - 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 + 12 T + 130 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 7 T + 75 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
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

−8.055654728816263958943295910872, −7.983632026711558776898517608607, −7.38285736491745071528538787897, −6.77682055724084887237329671925, −6.41165067596489630260515104553, −6.39207443516318198282613810262, −5.95634571640188433173166948228, −5.93741009904395761926143946770, −4.98346214175879970956748007345, −4.98335620458373946661440230188, −4.43479227425195452680727738060, −3.76114212618819706489735497142, −3.26541588307983695276272334365, −2.86840445691376213680526134083, −2.65269785413680882542054702615, −1.78526885550021860631648280189, −1.40621818432775280891407172359, −1.25733118263330262111971044933, 0, 0, 1.25733118263330262111971044933, 1.40621818432775280891407172359, 1.78526885550021860631648280189, 2.65269785413680882542054702615, 2.86840445691376213680526134083, 3.26541588307983695276272334365, 3.76114212618819706489735497142, 4.43479227425195452680727738060, 4.98335620458373946661440230188, 4.98346214175879970956748007345, 5.93741009904395761926143946770, 5.95634571640188433173166948228, 6.39207443516318198282613810262, 6.41165067596489630260515104553, 6.77682055724084887237329671925, 7.38285736491745071528538787897, 7.983632026711558776898517608607, 8.055654728816263958943295910872

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