Properties

Label 4-1134e2-1.1-c1e2-0-33
Degree $4$
Conductor $1285956$
Sign $-1$
Analytic cond. $81.9936$
Root an. cond. $3.00915$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 5·7-s − 9·13-s + 16-s + 3·19-s − 4·25-s + 5·28-s − 6·31-s + 2·37-s + 5·43-s + 18·49-s + 9·52-s + 24·61-s − 64-s − 5·67-s + 15·73-s − 3·76-s + 79-s + 45·91-s − 3·97-s + 4·100-s − 6·103-s − 13·109-s − 5·112-s − 2·121-s + 6·124-s + 127-s + ⋯
L(s)  = 1  − 1/2·4-s − 1.88·7-s − 2.49·13-s + 1/4·16-s + 0.688·19-s − 4/5·25-s + 0.944·28-s − 1.07·31-s + 0.328·37-s + 0.762·43-s + 18/7·49-s + 1.24·52-s + 3.07·61-s − 1/8·64-s − 0.610·67-s + 1.75·73-s − 0.344·76-s + 0.112·79-s + 4.71·91-s − 0.304·97-s + 2/5·100-s − 0.591·103-s − 1.24·109-s − 0.472·112-s − 0.181·121-s + 0.538·124-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1285956\)    =    \(2^{2} \cdot 3^{8} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(81.9936\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1285956,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + T^{2} \)
3 \( 1 \)
7$C_2$ \( 1 + 5 T + p T^{2} \)
good5$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} ) \)
23$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 118 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−7.67918371034966317341898739640, −7.29608215640852014938738524515, −6.88380319454323658351359926684, −6.67163933586349326953919726298, −5.89558908458079674080632563459, −5.54543966907262155729566369981, −5.21427216072771122642742868592, −4.62435568397508960024539676080, −4.01565633579559040588858246396, −3.64974973457130813398801330943, −3.05497654528752751281555583361, −2.51435450629834078345326389878, −2.09888124746657522409382749046, −0.73128835159777595123380153748, 0, 0.73128835159777595123380153748, 2.09888124746657522409382749046, 2.51435450629834078345326389878, 3.05497654528752751281555583361, 3.64974973457130813398801330943, 4.01565633579559040588858246396, 4.62435568397508960024539676080, 5.21427216072771122642742868592, 5.54543966907262155729566369981, 5.89558908458079674080632563459, 6.67163933586349326953919726298, 6.88380319454323658351359926684, 7.29608215640852014938738524515, 7.67918371034966317341898739640

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