Properties

Label 4-3360e2-1.1-c1e2-0-2
Degree $4$
Conductor $11289600$
Sign $1$
Analytic cond. $719.834$
Root an. cond. $5.17974$
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  − 2·3-s − 4·5-s + 3·9-s − 12·13-s + 8·15-s + 11·25-s − 4·27-s + 16·31-s − 4·37-s + 24·39-s − 16·41-s + 12·43-s − 12·45-s − 49-s − 12·53-s + 48·65-s + 4·67-s + 16·71-s − 22·75-s − 20·79-s + 5·81-s − 8·83-s − 32·93-s + 24·107-s + 8·111-s − 36·117-s + 22·121-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.78·5-s + 9-s − 3.32·13-s + 2.06·15-s + 11/5·25-s − 0.769·27-s + 2.87·31-s − 0.657·37-s + 3.84·39-s − 2.49·41-s + 1.82·43-s − 1.78·45-s − 1/7·49-s − 1.64·53-s + 5.95·65-s + 0.488·67-s + 1.89·71-s − 2.54·75-s − 2.25·79-s + 5/9·81-s − 0.878·83-s − 3.31·93-s + 2.32·107-s + 0.759·111-s − 3.32·117-s + 2·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11289600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11289600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(11289600\)    =    \(2^{10} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(719.834\)
Root analytic conductor: \(5.17974\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3360} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 11289600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3990053560\)
\(L(\frac12)\) \(\approx\) \(0.3990053560\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 4 T + p T^{2} \)
7$C_2$ \( 1 + T^{2} \)
good11$C_2$ \( ( 1 - p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 90 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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

−8.682811177184510242258009697753, −8.330930547511987971438085556120, −7.80531120093154186454057323951, −7.77841685468538486035892709426, −7.23329015721359118440342809506, −6.96567477981761374923079124435, −6.76772507938308963759707337596, −6.28699624342385626985278406297, −5.73419146841517276423694697831, −5.16868697802840566543630991686, −4.82702048847931642479359662314, −4.73957362977125156160147748847, −4.37000814100310825777023089899, −3.89895878746919370239977757652, −3.19149967919829601241504471599, −2.89560237545425143389815825411, −2.38518871903012722702557574717, −1.69732792631369941281081976029, −0.74833576553058149841603470573, −0.30728108029786993346128404027, 0.30728108029786993346128404027, 0.74833576553058149841603470573, 1.69732792631369941281081976029, 2.38518871903012722702557574717, 2.89560237545425143389815825411, 3.19149967919829601241504471599, 3.89895878746919370239977757652, 4.37000814100310825777023089899, 4.73957362977125156160147748847, 4.82702048847931642479359662314, 5.16868697802840566543630991686, 5.73419146841517276423694697831, 6.28699624342385626985278406297, 6.76772507938308963759707337596, 6.96567477981761374923079124435, 7.23329015721359118440342809506, 7.77841685468538486035892709426, 7.80531120093154186454057323951, 8.330930547511987971438085556120, 8.682811177184510242258009697753

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