Properties

Label 4-1148e2-1.1-c1e2-0-3
Degree $4$
Conductor $1317904$
Sign $-1$
Analytic cond. $84.0307$
Root an. cond. $3.02767$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s − 4·5-s − 3·8-s + 3·9-s − 4·10-s − 2·13-s − 16-s + 4·17-s + 3·18-s + 4·20-s + 3·25-s − 2·26-s − 2·29-s + 5·32-s + 4·34-s − 3·36-s + 4·37-s + 12·40-s − 12·41-s − 12·45-s − 49-s + 3·50-s + 2·52-s + 10·53-s − 2·58-s + 4·61-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 1.78·5-s − 1.06·8-s + 9-s − 1.26·10-s − 0.554·13-s − 1/4·16-s + 0.970·17-s + 0.707·18-s + 0.894·20-s + 3/5·25-s − 0.392·26-s − 0.371·29-s + 0.883·32-s + 0.685·34-s − 1/2·36-s + 0.657·37-s + 1.89·40-s − 1.87·41-s − 1.78·45-s − 1/7·49-s + 0.424·50-s + 0.277·52-s + 1.37·53-s − 0.262·58-s + 0.512·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1317904\)    =    \(2^{4} \cdot 7^{2} \cdot 41^{2}\)
Sign: $-1$
Analytic conductor: \(84.0307\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1317904} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1317904,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + p T^{2} \)
7$C_2$ \( 1 + T^{2} \)
41$C_2$ \( 1 + 12 T + p T^{2} \)
good3$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
17$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
31$C_2^2$ \( 1 - 27 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 55 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 39 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
59$C_2^2$ \( 1 + 115 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
67$C_2^2$ \( 1 + 97 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 73 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 89 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 91 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 8 T + p T^{2} ) \)
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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

−7.58991402882434767176800950547, −7.35392594568385346862560569400, −7.16996060519236524570317010242, −6.25732656422371547403268186266, −6.12117328897382878839267077346, −5.21302877583932413947222007031, −5.08281477433864113140272604013, −4.52199048449353000563317420658, −3.99893912434989633280304918243, −3.79469965445300721118601646658, −3.36423607000611598911854034589, −2.75305470475718881800757537055, −1.86633571095393736222123431096, −0.881482214760266314804723977236, 0, 0.881482214760266314804723977236, 1.86633571095393736222123431096, 2.75305470475718881800757537055, 3.36423607000611598911854034589, 3.79469965445300721118601646658, 3.99893912434989633280304918243, 4.52199048449353000563317420658, 5.08281477433864113140272604013, 5.21302877583932413947222007031, 6.12117328897382878839267077346, 6.25732656422371547403268186266, 7.16996060519236524570317010242, 7.35392594568385346862560569400, 7.58991402882434767176800950547

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