Properties

Label 4-3120e2-1.1-c1e2-0-20
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 $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s − 9-s + 12·11-s + 12·19-s + 11·25-s − 4·29-s − 8·31-s − 12·41-s + 4·45-s + 14·49-s − 48·55-s + 20·59-s − 12·61-s + 16·71-s + 32·79-s + 81-s + 20·89-s − 48·95-s − 12·99-s − 4·101-s + 32·109-s + 86·121-s − 24·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 1.78·5-s − 1/3·9-s + 3.61·11-s + 2.75·19-s + 11/5·25-s − 0.742·29-s − 1.43·31-s − 1.87·41-s + 0.596·45-s + 2·49-s − 6.47·55-s + 2.60·59-s − 1.53·61-s + 1.89·71-s + 3.60·79-s + 1/9·81-s + 2.11·89-s − 4.92·95-s − 1.20·99-s − 0.398·101-s + 3.06·109-s + 7.81·121-s − 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-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: induced by $\chi_{3120} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9734400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.947482231\)
\(L(\frac12)\) \(\approx\) \(2.947482231\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 + 4 T + p T^{2} \)
13$C_2$ \( 1 + T^{2} \)
good7$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
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$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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

−8.985638146542112474649593913011, −8.605584124902338579934079306047, −8.014898246931842699034920107393, −7.79859561663993858704704282312, −7.22072733554941409889066020828, −7.14499278852233543646786149229, −6.65812003953942996506590022210, −6.53827359966015120883962012832, −5.68929262885963082361920206186, −5.58868514554132514169985622503, −4.79248334589845639249736729994, −4.72049491964891259502320258345, −3.81001448163289466907465273624, −3.71383187391819415176085342724, −3.56158912243356156542751275670, −3.27989569536205576237731261848, −2.24150652283771042733532585058, −1.65176243207974130759187324214, −0.905334442637417059981850343670, −0.76724170859269516958330879521, 0.76724170859269516958330879521, 0.905334442637417059981850343670, 1.65176243207974130759187324214, 2.24150652283771042733532585058, 3.27989569536205576237731261848, 3.56158912243356156542751275670, 3.71383187391819415176085342724, 3.81001448163289466907465273624, 4.72049491964891259502320258345, 4.79248334589845639249736729994, 5.58868514554132514169985622503, 5.68929262885963082361920206186, 6.53827359966015120883962012832, 6.65812003953942996506590022210, 7.14499278852233543646786149229, 7.22072733554941409889066020828, 7.79859561663993858704704282312, 8.014898246931842699034920107393, 8.605584124902338579934079306047, 8.985638146542112474649593913011

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