Properties

Label 4-2394e2-1.1-c1e2-0-4
Degree $4$
Conductor $5731236$
Sign $1$
Analytic cond. $365.428$
Root an. cond. $4.37220$
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 + 2·7-s + 8-s + 3·10-s + 12·11-s − 5·13-s − 2·14-s − 16-s − 6·17-s − 19-s − 12·22-s − 3·23-s + 5·25-s + 5·26-s + 6·29-s − 8·31-s + 6·34-s − 6·35-s + 16·37-s + 38-s − 3·40-s + 6·41-s + 4·43-s + 3·46-s + 6·47-s + 3·49-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.34·5-s + 0.755·7-s + 0.353·8-s + 0.948·10-s + 3.61·11-s − 1.38·13-s − 0.534·14-s − 1/4·16-s − 1.45·17-s − 0.229·19-s − 2.55·22-s − 0.625·23-s + 25-s + 0.980·26-s + 1.11·29-s − 1.43·31-s + 1.02·34-s − 1.01·35-s + 2.63·37-s + 0.162·38-s − 0.474·40-s + 0.937·41-s + 0.609·43-s + 0.442·46-s + 0.875·47-s + 3/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5731236 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5731236 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5731236\)    =    \(2^{2} \cdot 3^{4} \cdot 7^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(365.428\)
Root analytic conductor: \(4.37220\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5731236,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.136494954\)
\(L(\frac12)\) \(\approx\) \(1.136494954\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
19$C_2$ \( 1 + T + p T^{2} \)
good5$C_2^2$ \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
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$C_2^2$ \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \)
67$C_2^2$ \( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 3 T - 62 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 4 T - 57 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
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

−9.065089087978926621678223299756, −8.877290747695455478508886466250, −8.478964609417662127624443359851, −8.088845259334253567837359767743, −7.59466234094809917145286280806, −7.17115200286137465120695092347, −7.14586737733098363587534372286, −6.53978376501671198090063981037, −6.26617674338902892488988018934, −5.74560766387340975041462990873, −5.12291520735608662543774538075, −4.32033307344482117162599873241, −4.30678440294243884799756986519, −4.08514119385663899794742989753, −3.83647788796639214552358228690, −2.64430404921242313233186325293, −2.56310454661527329649042056031, −1.49794307760171434164442726551, −1.29042999942740949366716676356, −0.45862839633879342325185375475, 0.45862839633879342325185375475, 1.29042999942740949366716676356, 1.49794307760171434164442726551, 2.56310454661527329649042056031, 2.64430404921242313233186325293, 3.83647788796639214552358228690, 4.08514119385663899794742989753, 4.30678440294243884799756986519, 4.32033307344482117162599873241, 5.12291520735608662543774538075, 5.74560766387340975041462990873, 6.26617674338902892488988018934, 6.53978376501671198090063981037, 7.14586737733098363587534372286, 7.17115200286137465120695092347, 7.59466234094809917145286280806, 8.088845259334253567837359767743, 8.478964609417662127624443359851, 8.877290747695455478508886466250, 9.065089087978926621678223299756

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