Properties

Label 4-1638e2-1.1-c1e2-0-36
Degree $4$
Conductor $2683044$
Sign $1$
Analytic cond. $171.073$
Root an. cond. $3.61655$
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  − 4-s + 4·13-s + 16-s + 4·17-s + 12·23-s + 6·25-s + 8·43-s − 49-s − 4·52-s − 8·53-s + 24·61-s − 64-s − 4·68-s − 12·92-s − 6·100-s − 4·101-s + 28·103-s + 24·107-s − 28·113-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  − 1/2·4-s + 1.10·13-s + 1/4·16-s + 0.970·17-s + 2.50·23-s + 6/5·25-s + 1.21·43-s − 1/7·49-s − 0.554·52-s − 1.09·53-s + 3.07·61-s − 1/8·64-s − 0.485·68-s − 1.25·92-s − 3/5·100-s − 0.398·101-s + 2.75·103-s + 2.32·107-s − 2.63·113-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2683044\)    =    \(2^{2} \cdot 3^{4} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(171.073\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1638} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2683044,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.859334456\)
\(L(\frac12)\) \(\approx\) \(2.859334456\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3 \( 1 \)
7$C_2$ \( 1 + T^{2} \)
13$C_2$ \( 1 - 4 T + p T^{2} \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 - p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 162 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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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

−9.424698992451957286159519272060, −9.152418986267061821224835424456, −8.749168164747724714042925181899, −8.595071690056888259044084140301, −7.934325375544616487882788847408, −7.75638654433286324095934684822, −7.06286264566425336447786157760, −6.84202525551738398982228498201, −6.45353718205156970366882323021, −5.77219638747517919045293616973, −5.52967677766559342397443899790, −5.04112035904256504823063621828, −4.66830019815387131361063796642, −4.16199777834455602780680036225, −3.50908506190115822744657955592, −3.24666959081167690420024510323, −2.76494907188270627237837438418, −1.97509705869079714685717712568, −0.977408154245149686882164570202, −0.924820628258119128540482393297, 0.924820628258119128540482393297, 0.977408154245149686882164570202, 1.97509705869079714685717712568, 2.76494907188270627237837438418, 3.24666959081167690420024510323, 3.50908506190115822744657955592, 4.16199777834455602780680036225, 4.66830019815387131361063796642, 5.04112035904256504823063621828, 5.52967677766559342397443899790, 5.77219638747517919045293616973, 6.45353718205156970366882323021, 6.84202525551738398982228498201, 7.06286264566425336447786157760, 7.75638654433286324095934684822, 7.934325375544616487882788847408, 8.595071690056888259044084140301, 8.749168164747724714042925181899, 9.152418986267061821224835424456, 9.424698992451957286159519272060

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