Properties

Label 4-2898e2-1.1-c1e2-0-2
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

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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 + 8·11-s + 5·13-s + 4·14-s + 5·16-s − 8·17-s + 2·19-s − 3·20-s + 16·22-s − 2·23-s + 25-s + 10·26-s + 6·28-s − 7·29-s − 4·31-s + 6·32-s − 16·34-s − 2·35-s + 15·37-s + 4·38-s − 4·40-s + 7·41-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 + 2.41·11-s + 1.38·13-s + 1.06·14-s + 5/4·16-s − 1.94·17-s + 0.458·19-s − 0.670·20-s + 3.41·22-s − 0.417·23-s + 1/5·25-s + 1.96·26-s + 1.13·28-s − 1.29·29-s − 0.718·31-s + 1.06·32-s − 2.74·34-s − 0.338·35-s + 2.46·37-s + 0.648·38-s − 0.632·40-s + 1.09·41-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

\(L(1)\) \(\approx\) \(10.06189890\)
\(L(\frac12)\) \(\approx\) \(10.06189890\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 - 5 T + 22 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
37$D_{4}$ \( 1 - 15 T + 120 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 7 T + 84 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 3 T + 78 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 6 T + 74 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 2 T + 78 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 - 4 T - 26 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 2 T + 102 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 10 T + 130 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$D_{4}$ \( 1 + 14 T + 174 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 10 T + 162 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 19 T + 274 T^{2} - 19 p T^{3} + p^{2} T^{4} \)
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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

−9.126757857835755877839062312135, −8.500622646010457550698242244657, −8.146652912397608881474640982753, −7.77559038018779645695992737940, −7.17492355305916466483826233031, −7.05225369316589681007727539576, −6.59469215707706262623068516039, −6.14073453144146507667082244266, −5.86800538294389149040174988243, −5.71794245741845818685227151327, −4.87896623063302869473157119088, −4.48266346907206762502895641965, −4.15045559389074793520254337776, −4.00730420632544595064044240466, −3.59550586560046086395018239392, −3.10296351108977347968222332467, −2.20284918099536179505614135847, −2.09490712769327407997606593046, −1.29393395565374479541145938594, −0.882849108044602374910707935437, 0.882849108044602374910707935437, 1.29393395565374479541145938594, 2.09490712769327407997606593046, 2.20284918099536179505614135847, 3.10296351108977347968222332467, 3.59550586560046086395018239392, 4.00730420632544595064044240466, 4.15045559389074793520254337776, 4.48266346907206762502895641965, 4.87896623063302869473157119088, 5.71794245741845818685227151327, 5.86800538294389149040174988243, 6.14073453144146507667082244266, 6.59469215707706262623068516039, 7.05225369316589681007727539576, 7.17492355305916466483826233031, 7.77559038018779645695992737940, 8.146652912397608881474640982753, 8.500622646010457550698242244657, 9.126757857835755877839062312135

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