Properties

Label 4-3920e2-1.1-c0e2-0-2
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  − 2·5-s + 9-s + 3·25-s − 2·29-s + 2·41-s − 2·45-s + 2·61-s − 2·89-s − 2·101-s + 2·109-s + 2·121-s − 4·125-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 + ⋯
L(s)  = 1  − 2·5-s + 9-s + 3·25-s − 2·29-s + 2·41-s − 2·45-s + 2·61-s − 2·89-s − 2·101-s + 2·109-s + 2·121-s − 4·125-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 + ⋯

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

\(L(\frac{1}{2})\) \(\approx\) \(0.9032356442\)
\(L(\frac12)\) \(\approx\) \(0.9032356442\)
\(L(1)\) 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$C_1$ \( ( 1 + T )^{2} \)
7 \( 1 \)
good3$C_2^2$ \( 1 - T^{2} + T^{4} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
23$C_2^2$ \( 1 - T^{2} + T^{4} \)
29$C_2$ \( ( 1 + T + T^{2} )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
41$C_2$ \( ( 1 - T + T^{2} )^{2} \)
43$C_2^2$ \( 1 - T^{2} + T^{4} \)
47$C_2$ \( ( 1 + T^{2} )^{2} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
83$C_2^2$ \( 1 - T^{2} + T^{4} \)
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

−8.670518571064718181213019787662, −8.468516270749958112318313506889, −7.948908863248564490121492483308, −7.76655971803882216531330490111, −7.35407386036474982897199637872, −7.06217667125121731686400338468, −6.92373550296238589167841172083, −6.38169813902501019082743582940, −5.68223021165370224851472751190, −5.59242626274693491723804348987, −4.95782050175129613019179083427, −4.47862804396335827529305299116, −4.21143610443135802774432277955, −3.94997644358720342591134005476, −3.52953135609114500770691446304, −3.10620335820860592271843950989, −2.54316683244857705306886928076, −1.95033197646336209607277874156, −1.25704503398957430265168342870, −0.59689074176784338880675773165, 0.59689074176784338880675773165, 1.25704503398957430265168342870, 1.95033197646336209607277874156, 2.54316683244857705306886928076, 3.10620335820860592271843950989, 3.52953135609114500770691446304, 3.94997644358720342591134005476, 4.21143610443135802774432277955, 4.47862804396335827529305299116, 4.95782050175129613019179083427, 5.59242626274693491723804348987, 5.68223021165370224851472751190, 6.38169813902501019082743582940, 6.92373550296238589167841172083, 7.06217667125121731686400338468, 7.35407386036474982897199637872, 7.76655971803882216531330490111, 7.948908863248564490121492483308, 8.468516270749958112318313506889, 8.670518571064718181213019787662

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