Properties

Label 4-1134e2-1.1-c1e2-0-31
Degree $4$
Conductor $1285956$
Sign $1$
Analytic cond. $81.9936$
Root an. cond. $3.00915$
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  + 4-s + 2·7-s + 4·13-s + 16-s − 14·19-s − 25-s + 2·28-s + 4·31-s + 4·37-s + 4·43-s + 3·49-s + 4·52-s + 10·61-s + 64-s + 16·67-s + 4·73-s − 14·76-s + 10·79-s + 8·91-s + 4·97-s − 100-s − 20·103-s − 20·109-s + 2·112-s + 14·121-s + 4·124-s + 127-s + ⋯
L(s)  = 1  + 1/2·4-s + 0.755·7-s + 1.10·13-s + 1/4·16-s − 3.21·19-s − 1/5·25-s + 0.377·28-s + 0.718·31-s + 0.657·37-s + 0.609·43-s + 3/7·49-s + 0.554·52-s + 1.28·61-s + 1/8·64-s + 1.95·67-s + 0.468·73-s − 1.60·76-s + 1.12·79-s + 0.838·91-s + 0.406·97-s − 0.0999·100-s − 1.97·103-s − 1.91·109-s + 0.188·112-s + 1.27·121-s + 0.359·124-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: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1285956,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.576348256\)
\(L(\frac12)\) \(\approx\) \(2.576348256\)
\(L(\frac{3}{2})\) 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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3 \( 1 \)
7$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−8.094785783071409988385687751137, −7.73532298603754652567815778453, −6.84669532198488939637919288494, −6.73871282898076223759796265441, −6.36293944731828035024321117773, −5.73900595462088482953892956618, −5.56957044429086118267228114847, −4.59790954336953877055233514712, −4.49364756499371052867692865672, −3.87247358034755873056669071066, −3.48588234639422265078311386338, −2.44868872871059860732796494913, −2.28395108895772497375168075701, −1.60097986545961424167062353389, −0.71201387828374154761208002119, 0.71201387828374154761208002119, 1.60097986545961424167062353389, 2.28395108895772497375168075701, 2.44868872871059860732796494913, 3.48588234639422265078311386338, 3.87247358034755873056669071066, 4.49364756499371052867692865672, 4.59790954336953877055233514712, 5.56957044429086118267228114847, 5.73900595462088482953892956618, 6.36293944731828035024321117773, 6.73871282898076223759796265441, 6.84669532198488939637919288494, 7.73532298603754652567815778453, 8.094785783071409988385687751137

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