Properties

Degree $4$
Conductor $36036009$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·4-s − 4·7-s + 2·13-s + 12·16-s − 2·17-s − 2·19-s + 2·23-s − 4·25-s + 16·28-s − 2·29-s − 16·31-s + 10·37-s + 4·41-s − 10·43-s + 20·47-s + 4·49-s − 8·52-s + 4·53-s + 6·59-s − 16·61-s − 32·64-s − 8·67-s + 8·68-s + 10·71-s − 4·73-s + 8·76-s + 2·79-s + ⋯
L(s)  = 1  − 2·4-s − 1.51·7-s + 0.554·13-s + 3·16-s − 0.485·17-s − 0.458·19-s + 0.417·23-s − 4/5·25-s + 3.02·28-s − 0.371·29-s − 2.87·31-s + 1.64·37-s + 0.624·41-s − 1.52·43-s + 2.91·47-s + 4/7·49-s − 1.10·52-s + 0.549·53-s + 0.781·59-s − 2.04·61-s − 4·64-s − 0.977·67-s + 0.970·68-s + 1.18·71-s − 0.468·73-s + 0.917·76-s + 0.225·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36036009 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36036009 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(36036009\)    =    \(3^{4} \cdot 23^{2} \cdot 29^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{6003} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 36036009,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad3 \( 1 \)
23$C_1$ \( ( 1 - T )^{2} \)
29$C_1$ \( ( 1 + T )^{2} \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 + 2 T + 29 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
31$D_{4}$ \( 1 + 16 T + 120 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 10 T + 75 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 4 T + 32 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 10 T + 87 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 20 T + 4 p T^{2} - 20 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 - 6 T + 121 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 16 T + 162 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 10 T + 113 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 4 T + 4 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 2 T + 135 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 20 T + 260 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 6 T + 181 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 24 T + 314 T^{2} - 24 p T^{3} + p^{2} T^{4} \)
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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

−7.897096936263980881127084495039, −7.57382637482461060688925434005, −7.30725985450537149508723027077, −6.90747393217435094851690655451, −6.22653860577511542911024161343, −6.11849428646037861499601062937, −5.74802905672627806791710638245, −5.46902720402960450624228798244, −4.95646128454021496260441118014, −4.62696712134164529345462278958, −4.11443465569389324016325618141, −3.83251270554898348963212431355, −3.57824233798080791223310775096, −3.31272655702183652625344964858, −2.59424369458930025707687990159, −2.18581143826740635576365838203, −1.38517596150965033471377151376, −0.862305715804540496833711459923, 0, 0, 0.862305715804540496833711459923, 1.38517596150965033471377151376, 2.18581143826740635576365838203, 2.59424369458930025707687990159, 3.31272655702183652625344964858, 3.57824233798080791223310775096, 3.83251270554898348963212431355, 4.11443465569389324016325618141, 4.62696712134164529345462278958, 4.95646128454021496260441118014, 5.46902720402960450624228798244, 5.74802905672627806791710638245, 6.11849428646037861499601062937, 6.22653860577511542911024161343, 6.90747393217435094851690655451, 7.30725985450537149508723027077, 7.57382637482461060688925434005, 7.897096936263980881127084495039

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