Properties

Label 4-3240e2-1.1-c1e2-0-4
Degree $4$
Conductor $10497600$
Sign $1$
Analytic cond. $669.336$
Root an. cond. $5.08640$
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  − 5-s + 4·7-s + 2·11-s − 4·13-s − 2·17-s − 10·19-s + 5·23-s + 8·29-s − 7·31-s − 4·35-s − 12·37-s + 6·41-s + 2·43-s + 8·47-s + 7·49-s − 18·53-s − 2·55-s + 4·59-s − 13·61-s + 4·65-s + 10·67-s + 12·71-s − 12·73-s + 8·77-s − 9·79-s − 17·83-s + 2·85-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.51·7-s + 0.603·11-s − 1.10·13-s − 0.485·17-s − 2.29·19-s + 1.04·23-s + 1.48·29-s − 1.25·31-s − 0.676·35-s − 1.97·37-s + 0.937·41-s + 0.304·43-s + 1.16·47-s + 49-s − 2.47·53-s − 0.269·55-s + 0.520·59-s − 1.66·61-s + 0.496·65-s + 1.22·67-s + 1.42·71-s − 1.40·73-s + 0.911·77-s − 1.01·79-s − 1.86·83-s + 0.216·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10497600\)    =    \(2^{6} \cdot 3^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(669.336\)
Root analytic conductor: \(5.08640\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 10497600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.398472310\)
\(L(\frac12)\) \(\approx\) \(1.398472310\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 + T + T^{2} \)
good7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2^2$ \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 5 T + 2 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 8 T + 35 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 2 T - 39 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 9 T + 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 17 T + 206 T^{2} + 17 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
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

−8.874672441891285683138363408087, −8.459835713836355600206200583950, −8.220243529571390131186645929268, −7.64159108066045231990549051785, −7.41966874323242114008345338954, −7.01467121302289410494195635032, −6.73282750453757397433917084721, −6.10205378366196811075483769524, −6.00645260401279448955405386647, −5.18922832595218440742070616850, −4.86674180337396369879507346984, −4.68262680522268959672663001683, −4.31328079181721841188198261513, −3.82663791915193161632532984768, −3.37923914257925414098751370188, −2.61028784156667144848012063859, −2.34094684624041533279095256116, −1.72338794199229094653573514545, −1.35266748513216560630302293880, −0.36002440305948838818549975707, 0.36002440305948838818549975707, 1.35266748513216560630302293880, 1.72338794199229094653573514545, 2.34094684624041533279095256116, 2.61028784156667144848012063859, 3.37923914257925414098751370188, 3.82663791915193161632532984768, 4.31328079181721841188198261513, 4.68262680522268959672663001683, 4.86674180337396369879507346984, 5.18922832595218440742070616850, 6.00645260401279448955405386647, 6.10205378366196811075483769524, 6.73282750453757397433917084721, 7.01467121302289410494195635032, 7.41966874323242114008345338954, 7.64159108066045231990549051785, 8.220243529571390131186645929268, 8.459835713836355600206200583950, 8.874672441891285683138363408087

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