Properties

Label 4-3920e2-1.1-c0e2-0-3
Degree $4$
Conductor $15366400$
Sign $1$
Analytic cond. $3.82724$
Root an. cond. $1.39869$
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 + 9-s + 4·29-s − 4·41-s + 45-s + 2·61-s − 2·89-s − 2·101-s + 2·109-s − 121-s − 125-s + 127-s + 131-s + 137-s + 139-s + 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 5-s + 9-s + 4·29-s − 4·41-s + 45-s + 2·61-s − 2·89-s − 2·101-s + 2·109-s − 121-s − 125-s + 127-s + 131-s + 137-s + 139-s + 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(15366400\)    =    \(2^{8} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(3.82724\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3920} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 15366400,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.126749570\)
\(L(\frac12)\) \(\approx\) \(2.126749570\)
\(L(1)\) 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 \( 1 \)
5$C_2$ \( 1 - T + T^{2} \)
7 \( 1 \)
good3$C_2^2$ \( 1 - T^{2} + T^{4} \)
11$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
17$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
19$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
23$C_2^2$ \( 1 - T^{2} + T^{4} \)
29$C_1$ \( ( 1 - T )^{4} \)
31$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
37$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
41$C_1$ \( ( 1 + T )^{4} \)
43$C_2$ \( ( 1 + T^{2} )^{2} \)
47$C_2^2$ \( 1 - T^{2} + T^{4} \)
53$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
59$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
61$C_2$ \( ( 1 - T + T^{2} )^{2} \)
67$C_2^2$ \( 1 - T^{2} + T^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
79$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
83$C_2$ \( ( 1 + T^{2} )^{2} \)
89$C_2$ \( ( 1 + T + T^{2} )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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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

−8.658324089163570759377829810949, −8.544904331268213053730215793229, −8.076981934288995250842476762173, −7.976548706184446865390362053358, −7.12687349369153388765452661726, −6.78459089254537512914890994565, −6.75185943142755218652918484211, −6.45892390804888911059095724372, −5.84483935684594269645001444564, −5.43929766255881672517468222856, −5.06647583653431159683539426337, −4.78245619302691252748903274452, −4.27086280084149204760307857238, −3.99719055258620313241446925148, −3.19217669200412904210876527403, −3.03745872317234705303229426957, −2.43850170423643638790939319817, −1.85906446390295164884588257946, −1.48812474695699962345333133290, −0.883798214941200687861064139717, 0.883798214941200687861064139717, 1.48812474695699962345333133290, 1.85906446390295164884588257946, 2.43850170423643638790939319817, 3.03745872317234705303229426957, 3.19217669200412904210876527403, 3.99719055258620313241446925148, 4.27086280084149204760307857238, 4.78245619302691252748903274452, 5.06647583653431159683539426337, 5.43929766255881672517468222856, 5.84483935684594269645001444564, 6.45892390804888911059095724372, 6.75185943142755218652918484211, 6.78459089254537512914890994565, 7.12687349369153388765452661726, 7.976548706184446865390362053358, 8.076981934288995250842476762173, 8.544904331268213053730215793229, 8.658324089163570759377829810949

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