Properties

Label 4-1140e2-1.1-c1e2-0-13
Degree $4$
Conductor $1299600$
Sign $-1$
Analytic cond. $82.8636$
Root an. cond. $3.01710$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s − 2·5-s − 3·8-s + 9-s − 2·10-s − 8·13-s − 16-s + 4·17-s + 18-s + 2·20-s + 3·25-s − 8·26-s + 8·29-s + 5·32-s + 4·34-s − 36-s + 6·40-s − 2·45-s − 10·49-s + 3·50-s + 8·52-s − 4·53-s + 8·58-s + 4·61-s + 7·64-s + 16·65-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 0.894·5-s − 1.06·8-s + 1/3·9-s − 0.632·10-s − 2.21·13-s − 1/4·16-s + 0.970·17-s + 0.235·18-s + 0.447·20-s + 3/5·25-s − 1.56·26-s + 1.48·29-s + 0.883·32-s + 0.685·34-s − 1/6·36-s + 0.948·40-s − 0.298·45-s − 1.42·49-s + 0.424·50-s + 1.10·52-s − 0.549·53-s + 1.05·58-s + 0.512·61-s + 7/8·64-s + 1.98·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1299600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1299600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1299600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(82.8636\)
Root analytic conductor: \(3.01710\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1299600} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1299600,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + p T^{2} \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_1$ \( ( 1 + T )^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
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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

−7.69451426376713427108997802036, −7.28034875915901512150776673175, −7.07341113368588561296966396700, −6.27483396251328001307537385761, −6.11724584120824685520629468061, −5.15467699824668629469430799137, −5.07872552787716309543367621064, −4.69308900565895276645745276376, −4.19187417186236300755390321977, −3.72176395363745906322596037322, −2.93342543120652714539614603200, −2.93269055521134332794719136670, −1.97779909070487265858566219387, −0.895818257201027131012503394679, 0, 0.895818257201027131012503394679, 1.97779909070487265858566219387, 2.93269055521134332794719136670, 2.93342543120652714539614603200, 3.72176395363745906322596037322, 4.19187417186236300755390321977, 4.69308900565895276645745276376, 5.07872552787716309543367621064, 5.15467699824668629469430799137, 6.11724584120824685520629468061, 6.27483396251328001307537385761, 7.07341113368588561296966396700, 7.28034875915901512150776673175, 7.69451426376713427108997802036

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