Properties

Label 4-560e2-1.1-c1e2-0-33
Degree $4$
Conductor $313600$
Sign $1$
Analytic cond. $19.9954$
Root an. cond. $2.11462$
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 + 3·9-s + 6·11-s + 4·13-s + 15-s + 6·17-s + 8·19-s − 5·21-s + 3·23-s + 8·27-s + 6·29-s + 2·31-s + 6·33-s − 5·35-s − 8·37-s + 4·39-s − 6·41-s − 10·43-s + 3·45-s + 18·49-s + 6·51-s − 12·53-s + 6·55-s + 8·57-s + 61-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 1.88·7-s + 9-s + 1.80·11-s + 1.10·13-s + 0.258·15-s + 1.45·17-s + 1.83·19-s − 1.09·21-s + 0.625·23-s + 1.53·27-s + 1.11·29-s + 0.359·31-s + 1.04·33-s − 0.845·35-s − 1.31·37-s + 0.640·39-s − 0.937·41-s − 1.52·43-s + 0.447·45-s + 18/7·49-s + 0.840·51-s − 1.64·53-s + 0.809·55-s + 1.05·57-s + 0.128·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(313600\)    =    \(2^{8} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(19.9954\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{560} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 313600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.888152887\)
\(L(\frac12)\) \(\approx\) \(2.888152887\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 - T + T^{2} \)
7$C_2$ \( 1 + 5 T + p T^{2} \)
good3$C_2^2$ \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - T + p T^{2} ) \)
23$C_2^2$ \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2^2$ \( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 3 T - 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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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

−10.68989361494849463274435359719, −10.46058670377058933316801826783, −9.786938663600683787192105588512, −9.786880525507366575165268719843, −9.301038899712288362920988552275, −8.998119376833668451673315465199, −8.363417278521026213757216681673, −8.021720856044280653995378443389, −7.12329146953734365226760007333, −6.88162324167860436496306637714, −6.55582359669875240872898298476, −6.17272885613011890655167590188, −5.45673934731770641113599573184, −4.99720724392104398638989585932, −4.07725842477931560750023000586, −3.63128004650993655204502494964, −3.15177141816932700949011503777, −2.94219701204365767507581915833, −1.32923166829179909689328365356, −1.27941598824344767311977059718, 1.27941598824344767311977059718, 1.32923166829179909689328365356, 2.94219701204365767507581915833, 3.15177141816932700949011503777, 3.63128004650993655204502494964, 4.07725842477931560750023000586, 4.99720724392104398638989585932, 5.45673934731770641113599573184, 6.17272885613011890655167590188, 6.55582359669875240872898298476, 6.88162324167860436496306637714, 7.12329146953734365226760007333, 8.021720856044280653995378443389, 8.363417278521026213757216681673, 8.998119376833668451673315465199, 9.301038899712288362920988552275, 9.786880525507366575165268719843, 9.786938663600683787192105588512, 10.46058670377058933316801826783, 10.68989361494849463274435359719

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