Properties

Label 4-2898e2-1.1-c1e2-0-3
Degree $4$
Conductor $8398404$
Sign $1$
Analytic cond. $535.489$
Root an. cond. $4.81047$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·4-s + 2·5-s − 2·7-s + 4·8-s + 4·10-s − 4·13-s − 4·14-s + 5·16-s + 10·17-s + 6·20-s + 2·23-s − 2·25-s − 8·26-s − 6·28-s + 4·29-s + 2·31-s + 6·32-s + 20·34-s − 4·35-s + 12·37-s + 8·40-s + 20·41-s − 4·43-s + 4·46-s + 18·47-s + 3·49-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 0.894·5-s − 0.755·7-s + 1.41·8-s + 1.26·10-s − 1.10·13-s − 1.06·14-s + 5/4·16-s + 2.42·17-s + 1.34·20-s + 0.417·23-s − 2/5·25-s − 1.56·26-s − 1.13·28-s + 0.742·29-s + 0.359·31-s + 1.06·32-s + 3.42·34-s − 0.676·35-s + 1.97·37-s + 1.26·40-s + 3.12·41-s − 0.609·43-s + 0.589·46-s + 2.62·47-s + 3/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8398404 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8398404 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8398404\)    =    \(2^{2} \cdot 3^{4} \cdot 7^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(535.489\)
Root analytic conductor: \(4.81047\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 8398404,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(9.829853157\)
\(L(\frac12)\) \(\approx\) \(9.829853157\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
3 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
23$C_1$ \( ( 1 - T )^{2} \)
good5$D_{4}$ \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$D_{4}$ \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 10 T + 54 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$D_{4}$ \( 1 - 2 T + 58 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$D_{4}$ \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 18 T + 170 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 + 18 T + 194 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 10 T + 102 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_4$ \( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 10 T + 78 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 6 T + 78 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
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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

−9.011232051352668053479860286714, −8.636839045271403836950397545377, −7.85317617092382484549425487236, −7.69866297603610462479234142356, −7.30536837138067229526476926749, −7.20095061210415025914491298722, −6.26682397318652874660922930294, −6.23633993338354679934610087836, −5.82772073873977727731450362952, −5.62739557677466252773303188339, −5.07368544097290215752507173254, −4.79531071639154871394350040068, −4.14663407077729232926573451558, −3.90390297008214384071129987301, −3.37219491585996051010548066023, −2.77455627253228564364406023046, −2.49038445678904597617129358442, −2.29764599727947336505987216140, −1.13016948224013605500194388943, −0.921227657153810221553143388966, 0.921227657153810221553143388966, 1.13016948224013605500194388943, 2.29764599727947336505987216140, 2.49038445678904597617129358442, 2.77455627253228564364406023046, 3.37219491585996051010548066023, 3.90390297008214384071129987301, 4.14663407077729232926573451558, 4.79531071639154871394350040068, 5.07368544097290215752507173254, 5.62739557677466252773303188339, 5.82772073873977727731450362952, 6.23633993338354679934610087836, 6.26682397318652874660922930294, 7.20095061210415025914491298722, 7.30536837138067229526476926749, 7.69866297603610462479234142356, 7.85317617092382484549425487236, 8.636839045271403836950397545377, 9.011232051352668053479860286714

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