Properties

Label 4-840e2-1.1-c1e2-0-0
Degree $4$
Conductor $705600$
Sign $1$
Analytic cond. $44.9896$
Root an. cond. $2.58987$
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  + 3-s − 5-s − 5·7-s − 2·13-s − 15-s − 2·17-s − 3·19-s − 5·21-s − 6·23-s − 27-s − 12·29-s − 7·31-s + 5·35-s − 37-s − 2·39-s + 8·41-s − 22·43-s − 6·47-s + 18·49-s − 2·51-s + 4·53-s − 3·57-s + 14·59-s + 6·61-s + 2·65-s − 3·67-s − 6·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 1.88·7-s − 0.554·13-s − 0.258·15-s − 0.485·17-s − 0.688·19-s − 1.09·21-s − 1.25·23-s − 0.192·27-s − 2.22·29-s − 1.25·31-s + 0.845·35-s − 0.164·37-s − 0.320·39-s + 1.24·41-s − 3.35·43-s − 0.875·47-s + 18/7·49-s − 0.280·51-s + 0.549·53-s − 0.397·57-s + 1.82·59-s + 0.768·61-s + 0.248·65-s − 0.366·67-s − 0.722·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(705600\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(44.9896\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 705600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3541492181\)
\(L(\frac12)\) \(\approx\) \(0.3541492181\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( 1 - T + T^{2} \)
5$C_2$ \( 1 + T + T^{2} \)
7$C_2$ \( 1 + 5 T + p T^{2} \)
good11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 3 T - 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 14 T + 137 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
79$C_2^2$ \( 1 + 9 T + 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 2 T - 85 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
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

−10.45510241195139830220162722717, −9.742374957572972923284070334114, −9.668021520340069476844823891112, −9.178579906113245336376245507091, −8.859970446049462453371827149784, −8.186655272050686646242597547918, −8.045456363872615063648860959625, −7.25640185713538769946475006249, −7.08864664594521490616455662326, −6.55726200879393971943646292365, −6.21443793368941027434304273153, −5.42282850079940975749822787779, −5.38994211141245851590722255976, −4.21187681129792619017521654915, −4.06416568046873071713891536109, −3.32072436859445483115810099897, −3.25880862036106100342164937360, −2.18697747805290813565542580016, −1.98097094151156610271456291855, −0.26034704232310798578586493417, 0.26034704232310798578586493417, 1.98097094151156610271456291855, 2.18697747805290813565542580016, 3.25880862036106100342164937360, 3.32072436859445483115810099897, 4.06416568046873071713891536109, 4.21187681129792619017521654915, 5.38994211141245851590722255976, 5.42282850079940975749822787779, 6.21443793368941027434304273153, 6.55726200879393971943646292365, 7.08864664594521490616455662326, 7.25640185713538769946475006249, 8.045456363872615063648860959625, 8.186655272050686646242597547918, 8.859970446049462453371827149784, 9.178579906113245336376245507091, 9.668021520340069476844823891112, 9.742374957572972923284070334114, 10.45510241195139830220162722717

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