Properties

Label 4-1134e2-1.1-c1e2-0-11
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 − 3·5-s − 7-s + 8-s + 3·10-s + 3·11-s + 4·13-s + 14-s − 16-s + 12·17-s − 14·19-s − 3·22-s − 3·23-s + 5·25-s − 4·26-s − 5·31-s − 12·34-s + 3·35-s − 14·37-s + 14·38-s − 3·40-s − 9·41-s + 10·43-s + 3·46-s + 6·47-s − 5·50-s − 24·53-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.34·5-s − 0.377·7-s + 0.353·8-s + 0.948·10-s + 0.904·11-s + 1.10·13-s + 0.267·14-s − 1/4·16-s + 2.91·17-s − 3.21·19-s − 0.639·22-s − 0.625·23-s + 25-s − 0.784·26-s − 0.898·31-s − 2.05·34-s + 0.507·35-s − 2.30·37-s + 2.27·38-s − 0.474·40-s − 1.40·41-s + 1.52·43-s + 0.442·46-s + 0.875·47-s − 0.707·50-s − 3.29·53-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

\(L(1)\) \(\approx\) \(0.4795576985\)
\(L(\frac12)\) \(\approx\) \(0.4795576985\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
good5$C_2^2$ \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 8 T - 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
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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

−10.21307271205712844094756177272, −9.468783435582933890938923726914, −9.180688730585638236743667316358, −8.682221572666884403664101095894, −8.277300945373504316623385076704, −8.220710847916428281024307274536, −7.68730105763419994096189796142, −7.15597434769857276055366323903, −6.89048029226982556013977877695, −6.18746122893249894922842204710, −5.99419603552723828146712530956, −5.47568081231923783502518735379, −4.57150825542166129561227477087, −4.37128042795252183440702999653, −3.65195907699735958839635850664, −3.55058116213267723387501508738, −3.00902470852328674728299248161, −1.75087567053039235246962591562, −1.50210857448012576323703618716, −0.35845510065607159860200186987, 0.35845510065607159860200186987, 1.50210857448012576323703618716, 1.75087567053039235246962591562, 3.00902470852328674728299248161, 3.55058116213267723387501508738, 3.65195907699735958839635850664, 4.37128042795252183440702999653, 4.57150825542166129561227477087, 5.47568081231923783502518735379, 5.99419603552723828146712530956, 6.18746122893249894922842204710, 6.89048029226982556013977877695, 7.15597434769857276055366323903, 7.68730105763419994096189796142, 8.220710847916428281024307274536, 8.277300945373504316623385076704, 8.682221572666884403664101095894, 9.180688730585638236743667316358, 9.468783435582933890938923726914, 10.21307271205712844094756177272

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