Properties

Label 4-2898e2-1.1-c1e2-0-1
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 + 5-s − 2·7-s + 4·8-s + 2·10-s + 5·13-s − 4·14-s + 5·16-s + 2·19-s + 3·20-s + 2·23-s + 25-s + 10·26-s − 6·28-s − 7·29-s + 12·31-s + 6·32-s − 2·35-s + 37-s + 4·38-s + 4·40-s − 9·41-s + 5·43-s + 4·46-s − 47-s + 3·49-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 0.447·5-s − 0.755·7-s + 1.41·8-s + 0.632·10-s + 1.38·13-s − 1.06·14-s + 5/4·16-s + 0.458·19-s + 0.670·20-s + 0.417·23-s + 1/5·25-s + 1.96·26-s − 1.13·28-s − 1.29·29-s + 2.15·31-s + 1.06·32-s − 0.338·35-s + 0.164·37-s + 0.648·38-s + 0.632·40-s − 1.40·41-s + 0.762·43-s + 0.589·46-s − 0.145·47-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: induced by $\chi_{2898} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 8398404,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(8.798750732\)
\(L(\frac12)\) \(\approx\) \(8.798750732\)
\(L(\frac{3}{2})\) 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$D_{4}$ \( 1 - T - p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$D_{4}$ \( 1 - 5 T + 22 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$D_{4}$ \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 7 T + 60 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
37$D_{4}$ \( 1 - T + 64 T^{2} - p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 9 T + 92 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 5 T + 82 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + T + 84 T^{2} + p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 6 T + 74 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 14 T + 126 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 + 2 T + 102 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 2 T + 106 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$D_{4}$ \( 1 - 18 T + 206 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 18 T + 218 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 17 T + 174 T^{2} - 17 p T^{3} + p^{2} T^{4} \)
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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

−8.808923785932214279122683475355, −8.629440024529208531499535649140, −8.118161817542072837369374345554, −7.77497374755217167412557347728, −7.29231799063202544499763173212, −6.77536911204451311084019373597, −6.55299686641013405236612339592, −6.31825768553830574372918017727, −5.85857796891909436770674701139, −5.50479754903119785543483755890, −4.99071684813130730745196131379, −4.89685361524396843432395793535, −4.04923674355763669857687043844, −3.80688298153146717399058336994, −3.45527438155497129447700702379, −3.06243917093988772798819173715, −2.35857044104519559859925289303, −2.17680477337606366026284560933, −1.26143318420387332138780214808, −0.802794875645493798539687683775, 0.802794875645493798539687683775, 1.26143318420387332138780214808, 2.17680477337606366026284560933, 2.35857044104519559859925289303, 3.06243917093988772798819173715, 3.45527438155497129447700702379, 3.80688298153146717399058336994, 4.04923674355763669857687043844, 4.89685361524396843432395793535, 4.99071684813130730745196131379, 5.50479754903119785543483755890, 5.85857796891909436770674701139, 6.31825768553830574372918017727, 6.55299686641013405236612339592, 6.77536911204451311084019373597, 7.29231799063202544499763173212, 7.77497374755217167412557347728, 8.118161817542072837369374345554, 8.629440024529208531499535649140, 8.808923785932214279122683475355

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