Properties

Label 4-1134e2-1.1-c1e2-0-51
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 + 6·5-s − 4·7-s − 8-s + 6·10-s + 6·11-s + 4·13-s − 4·14-s − 16-s + 4·19-s + 6·22-s + 17·25-s + 4·26-s − 9·29-s + 31-s − 24·35-s − 8·37-s + 4·38-s − 6·40-s + 10·43-s + 6·47-s + 9·49-s + 17·50-s + 3·53-s + 36·55-s + 4·56-s − 9·58-s + ⋯
L(s)  = 1  + 0.707·2-s + 2.68·5-s − 1.51·7-s − 0.353·8-s + 1.89·10-s + 1.80·11-s + 1.10·13-s − 1.06·14-s − 1/4·16-s + 0.917·19-s + 1.27·22-s + 17/5·25-s + 0.784·26-s − 1.67·29-s + 0.179·31-s − 4.05·35-s − 1.31·37-s + 0.648·38-s − 0.948·40-s + 1.52·43-s + 0.875·47-s + 9/7·49-s + 2.40·50-s + 0.412·53-s + 4.85·55-s + 0.534·56-s − 1.18·58-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

\(L(1)\) \(\approx\) \(5.336760248\)
\(L(\frac12)\) \(\approx\) \(5.336760248\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 4 T + p T^{2} \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2^2$ \( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
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$C_2^2$ \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 10 T + 33 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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$C_2^2$ \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - T - 96 T^{2} - p T^{3} + p^{2} T^{4} \)
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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

−9.708486461572133428219338457256, −9.691122025135894266960511988428, −9.226637517113060471890790531188, −9.114712235258629449525666312509, −8.739549406789580451893183727713, −8.024947940780678758494062239078, −7.15498269762732117768440445654, −6.85418599755038843685713188580, −6.57932104836268738520748312337, −5.98305633488629024911144092968, −5.89224375491954948816210244283, −5.54156348627222818506458121028, −5.17560042390610385257426363034, −4.10870492650767046906267084903, −3.95234891995927076790138491199, −3.35661054123594239836902458680, −2.85087939988167470254451959724, −2.14775661198896399236058700745, −1.62767528143009047053710113133, −0.977203350482848606751121787607, 0.977203350482848606751121787607, 1.62767528143009047053710113133, 2.14775661198896399236058700745, 2.85087939988167470254451959724, 3.35661054123594239836902458680, 3.95234891995927076790138491199, 4.10870492650767046906267084903, 5.17560042390610385257426363034, 5.54156348627222818506458121028, 5.89224375491954948816210244283, 5.98305633488629024911144092968, 6.57932104836268738520748312337, 6.85418599755038843685713188580, 7.15498269762732117768440445654, 8.024947940780678758494062239078, 8.739549406789580451893183727713, 9.114712235258629449525666312509, 9.226637517113060471890790531188, 9.691122025135894266960511988428, 9.708486461572133428219338457256

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