Properties

Label 4-1092e2-1.1-c1e2-0-41
Degree $4$
Conductor $1192464$
Sign $-1$
Analytic cond. $76.0325$
Root an. cond. $2.95290$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·7-s + 9-s − 10·25-s − 12·29-s + 4·37-s − 8·43-s − 3·49-s + 12·53-s + 2·63-s − 20·67-s + 24·71-s + 16·79-s + 81-s − 24·107-s + 28·109-s + 12·113-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + ⋯
L(s)  = 1  + 0.755·7-s + 1/3·9-s − 2·25-s − 2.22·29-s + 0.657·37-s − 1.21·43-s − 3/7·49-s + 1.64·53-s + 0.251·63-s − 2.44·67-s + 2.84·71-s + 1.80·79-s + 1/9·81-s − 2.32·107-s + 2.68·109-s + 1.12·113-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1/13·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1192464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1192464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1192464\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(76.0325\)
Root analytic conductor: \(2.95290\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1192464} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1192464,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_2$ \( 1 - 2 T + p T^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−7.86871640882441783507560416572, −7.63751922966226696428333146989, −6.84970961425661294350595833560, −6.65843891177652373785895186115, −5.80908037906310159495222835771, −5.74410266320392890700491858552, −5.11454570698520102977754794044, −4.70428189754287337615499396994, −3.99747189532817479880871482610, −3.80067854107620357152276821608, −3.19862495704087502707534439075, −2.15249025710140806520166174974, −2.03881758362108754018941589683, −1.20293849450464588381831778675, 0, 1.20293849450464588381831778675, 2.03881758362108754018941589683, 2.15249025710140806520166174974, 3.19862495704087502707534439075, 3.80067854107620357152276821608, 3.99747189532817479880871482610, 4.70428189754287337615499396994, 5.11454570698520102977754794044, 5.74410266320392890700491858552, 5.80908037906310159495222835771, 6.65843891177652373785895186115, 6.84970961425661294350595833560, 7.63751922966226696428333146989, 7.86871640882441783507560416572

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