Properties

Label 4-1134e2-1.1-c1e2-0-7
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 5·7-s + 8-s − 12·11-s − 5·13-s − 5·14-s − 16-s + 6·17-s + 4·19-s + 12·22-s − 12·23-s − 10·25-s + 5·26-s + 6·29-s + 31-s − 6·34-s + 37-s − 4·38-s − 6·41-s + 43-s + 12·46-s − 6·47-s + 18·49-s + 10·50-s − 6·53-s + 5·56-s − 6·58-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.88·7-s + 0.353·8-s − 3.61·11-s − 1.38·13-s − 1.33·14-s − 1/4·16-s + 1.45·17-s + 0.917·19-s + 2.55·22-s − 2.50·23-s − 2·25-s + 0.980·26-s + 1.11·29-s + 0.179·31-s − 1.02·34-s + 0.164·37-s − 0.648·38-s − 0.937·41-s + 0.152·43-s + 1.76·46-s − 0.875·47-s + 18/7·49-s + 1.41·50-s − 0.824·53-s + 0.668·56-s − 0.787·58-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: induced by $\chi_{1134} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1285956,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4268110236\)
\(L(\frac12)\) \(\approx\) \(0.4268110236\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
3 \( 1 \)
7$C_2$ \( 1 - 5 T + p T^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2^2$ \( 1 - T - 66 T^{2} - p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 6 T - 47 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 17 T + 192 T^{2} + 17 p T^{3} + p^{2} T^{4} \)
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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

−9.958551526223595855206332710371, −9.800126945367223941535730406455, −9.460546799018121678264758689516, −8.302615108749417880550665931833, −8.242833893349697339424528780741, −8.008131415707988281563912985988, −7.83662499241586706486499338570, −7.40282711257185787720197550572, −7.13317487854057583238249128023, −5.88254054680190165166433383901, −5.76931195722910655549769074273, −5.16058731088957680612920971133, −5.12998615067236608923242002615, −4.48867585967093749663461055931, −4.07328144093397480858179774440, −2.87934063903172108645609724631, −2.84055454104471899862961679408, −1.93404512240286852963362138812, −1.68819024478826595079496178551, −0.30682151304072432538144532853, 0.30682151304072432538144532853, 1.68819024478826595079496178551, 1.93404512240286852963362138812, 2.84055454104471899862961679408, 2.87934063903172108645609724631, 4.07328144093397480858179774440, 4.48867585967093749663461055931, 5.12998615067236608923242002615, 5.16058731088957680612920971133, 5.76931195722910655549769074273, 5.88254054680190165166433383901, 7.13317487854057583238249128023, 7.40282711257185787720197550572, 7.83662499241586706486499338570, 8.008131415707988281563912985988, 8.242833893349697339424528780741, 8.302615108749417880550665931833, 9.460546799018121678264758689516, 9.800126945367223941535730406455, 9.958551526223595855206332710371

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