Properties

Label 4-3420e2-1.1-c0e2-0-1
Degree $4$
Conductor $11696400$
Sign $1$
Analytic cond. $2.91317$
Root an. cond. $1.30644$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 2·11-s + 2·19-s − 49-s − 2·55-s + 2·61-s + 2·95-s + 4·101-s + 121-s − 125-s + ⋯
L(s)  = 1  + 5-s − 2·11-s + 2·19-s − 49-s − 2·55-s + 2·61-s + 2·95-s + 4·101-s + 121-s − 125-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11696400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11696400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(11696400\)    =    \(2^{4} \cdot 3^{4} \cdot 5^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(2.91317\)
Root analytic conductor: \(1.30644\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 11696400,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.514182543\)
\(L(\frac12)\) \(\approx\) \(1.514182543\)
\(L(1)\) 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 \( 1 \)
5$C_2$ \( 1 - T + T^{2} \)
19$C_1$ \( ( 1 - T )^{2} \)
good7$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
11$C_2$ \( ( 1 + T + T^{2} )^{2} \)
13$C_2$ \( ( 1 + T^{2} )^{2} \)
17$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_2$ \( ( 1 + T^{2} )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
47$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
53$C_2$ \( ( 1 + T^{2} )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_2$ \( ( 1 - T + T^{2} )^{2} \)
67$C_2$ \( ( 1 + T^{2} )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
97$C_2$ \( ( 1 + T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.905574477027714282677781675335, −8.619621629813422481848007233415, −8.206662451910212620607517499213, −7.76636696498241087303353077965, −7.50489465557566279899137031451, −7.28555396692391030310139265831, −6.71034339543966271381691718488, −6.29234366362680868411565906656, −5.86890042022152039864705031678, −5.51458192167458795804042464448, −5.22971988558391834932662769370, −4.96399154522782779481482275731, −4.52890010262956596314025892402, −3.83739183333108917180272934079, −3.21657599996692471392800654820, −3.13311005417323666772238798720, −2.35810445145466476024325132745, −2.20634282294789445149447807980, −1.49265007423178537962110452318, −0.73993602968990485844533605341, 0.73993602968990485844533605341, 1.49265007423178537962110452318, 2.20634282294789445149447807980, 2.35810445145466476024325132745, 3.13311005417323666772238798720, 3.21657599996692471392800654820, 3.83739183333108917180272934079, 4.52890010262956596314025892402, 4.96399154522782779481482275731, 5.22971988558391834932662769370, 5.51458192167458795804042464448, 5.86890042022152039864705031678, 6.29234366362680868411565906656, 6.71034339543966271381691718488, 7.28555396692391030310139265831, 7.50489465557566279899137031451, 7.76636696498241087303353077965, 8.206662451910212620607517499213, 8.619621629813422481848007233415, 8.905574477027714282677781675335

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