Properties

Label 4-1216e2-1.1-c1e2-0-6
Degree $4$
Conductor $1478656$
Sign $-1$
Analytic cond. $94.2803$
Root an. cond. $3.11605$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·5-s − 6·9-s + 8·13-s − 6·17-s + 17·25-s − 16·37-s + 36·45-s + 11·49-s + 12·53-s + 10·61-s − 48·65-s − 22·73-s + 27·81-s + 36·85-s − 20·89-s − 24·97-s + 4·101-s − 12·109-s − 8·113-s − 48·117-s + 3·121-s − 18·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + ⋯
L(s)  = 1  − 2.68·5-s − 2·9-s + 2.21·13-s − 1.45·17-s + 17/5·25-s − 2.63·37-s + 5.36·45-s + 11/7·49-s + 1.64·53-s + 1.28·61-s − 5.95·65-s − 2.57·73-s + 3·81-s + 3.90·85-s − 2.11·89-s − 2.43·97-s + 0.398·101-s − 1.14·109-s − 0.752·113-s − 4.43·117-s + 3/11·121-s − 1.60·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1478656\)    =    \(2^{12} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(94.2803\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1478656} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1478656,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
19$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
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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

−7.86378097245535978247876093758, −7.18059484484420096366454394254, −6.98286792123187236566096654142, −6.56370906167522668644909842214, −5.76283520417985127913399041730, −5.67103421999051188975454804232, −5.05210681704737714861605588166, −4.14385266867940367577820086585, −4.09350585892641493698723914863, −3.71418757784612877583046899350, −3.10630100563499367104455654686, −2.78320977751625677782435169813, −1.77861406991832868392707836598, −0.65261125406809044714433414915, 0, 0.65261125406809044714433414915, 1.77861406991832868392707836598, 2.78320977751625677782435169813, 3.10630100563499367104455654686, 3.71418757784612877583046899350, 4.09350585892641493698723914863, 4.14385266867940367577820086585, 5.05210681704737714861605588166, 5.67103421999051188975454804232, 5.76283520417985127913399041730, 6.56370906167522668644909842214, 6.98286792123187236566096654142, 7.18059484484420096366454394254, 7.86378097245535978247876093758

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