Properties

Label 4-6e2-1.1-c6e2-0-0
Degree $4$
Conductor $36$
Sign $1$
Analytic cond. $1.90529$
Root an. cond. $1.17487$
Motivic weight $6$
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  + 42·3-s − 32·4-s + 4·7-s + 1.03e3·9-s − 1.34e3·12-s − 5.90e3·13-s + 1.02e3·16-s + 1.05e4·19-s + 168·21-s + 2.45e3·25-s + 1.28e4·27-s − 128·28-s + 4.57e4·31-s − 3.31e4·36-s + 6.81e4·37-s − 2.47e5·39-s − 1.28e4·43-s + 4.30e4·48-s − 2.35e5·49-s + 1.88e5·52-s + 4.41e5·57-s − 1.25e5·61-s + 4.14e3·63-s − 3.27e4·64-s + 8.77e5·67-s − 1.46e6·73-s + 1.02e5·75-s + ⋯
L(s)  = 1  + 14/9·3-s − 1/2·4-s + 0.0116·7-s + 1.41·9-s − 7/9·12-s − 2.68·13-s + 1/4·16-s + 1.53·19-s + 0.0181·21-s + 0.156·25-s + 0.652·27-s − 0.00583·28-s + 1.53·31-s − 0.709·36-s + 1.34·37-s − 4.17·39-s − 0.161·43-s + 7/18·48-s − 1.99·49-s + 1.34·52-s + 2.38·57-s − 0.551·61-s + 0.0165·63-s − 1/8·64-s + 2.91·67-s − 3.75·73-s + 0.243·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(1.90529\)
Root analytic conductor: \(1.17487\)
Motivic weight: \(6\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{6} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 36,\ (\ :3, 3),\ 1)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.613668613\)
\(L(\frac12)\) \(\approx\) \(1.613668613\)
\(L(4)\) 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$C_2$ \( 1 + p^{5} T^{2} \)
3$C_2$ \( 1 - 14 p T + p^{6} T^{2} \)
good5$C_2^2$ \( 1 - 98 p^{2} T^{2} + p^{12} T^{4} \)
7$C_2$ \( ( 1 - 2 T + p^{6} T^{2} )^{2} \)
11$C_2^2$ \( 1 - 3541970 T^{2} + p^{12} T^{4} \)
13$C_2$ \( ( 1 + 2950 T + p^{6} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 28202690 T^{2} + p^{12} T^{4} \)
19$C_2$ \( ( 1 - 5258 T + p^{6} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 191004770 T^{2} + p^{12} T^{4} \)
29$C_2^2$ \( 1 - 1184779442 T^{2} + p^{12} T^{4} \)
31$C_2$ \( ( 1 - 22898 T + p^{6} T^{2} )^{2} \)
37$C_2$ \( ( 1 - 34058 T + p^{6} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 9219079010 T^{2} + p^{12} T^{4} \)
43$C_2$ \( ( 1 + 6406 T + p^{6} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 10801249342 T^{2} + p^{12} T^{4} \)
53$C_2^2$ \( 1 - 7253988050 T^{2} + p^{12} T^{4} \)
59$C_2^2$ \( 1 + 22449655150 T^{2} + p^{12} T^{4} \)
61$C_2$ \( ( 1 + 62566 T + p^{6} T^{2} )^{2} \)
67$C_2$ \( ( 1 - 438698 T + p^{6} T^{2} )^{2} \)
71$C_2^2$ \( 1 - 251546372642 T^{2} + p^{12} T^{4} \)
73$C_2$ \( ( 1 + 730510 T + p^{6} T^{2} )^{2} \)
79$C_2$ \( ( 1 - 340562 T + p^{6} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 407613512306 T^{2} + p^{12} T^{4} \)
89$C_2^2$ \( 1 - 844406214050 T^{2} + p^{12} T^{4} \)
97$C_2$ \( ( 1 + 281086 T + p^{6} T^{2} )^{2} \)
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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

−22.13373863258863335150093069922, −21.75047343802629783924738298655, −20.72176959406186343523486861247, −20.05673684917198431363307465258, −19.46638715118433574189026732750, −18.93848836651635313512249730379, −17.87641256636620115997628881121, −17.12696437855432872860112574169, −15.99713022351561678227859458465, −14.91387849581282328263842151229, −14.49119320717986983194720810727, −13.73957209498856935585648521689, −12.78562237311126828980232507898, −11.81246219455376496576037678043, −9.782251679734233016659644703039, −9.617620391103364735432910615057, −8.159996850570753693766998243183, −7.32356439514048219557212455775, −4.78440219298001570693256558400, −2.81060547486885503373252387275, 2.81060547486885503373252387275, 4.78440219298001570693256558400, 7.32356439514048219557212455775, 8.159996850570753693766998243183, 9.617620391103364735432910615057, 9.782251679734233016659644703039, 11.81246219455376496576037678043, 12.78562237311126828980232507898, 13.73957209498856935585648521689, 14.49119320717986983194720810727, 14.91387849581282328263842151229, 15.99713022351561678227859458465, 17.12696437855432872860112574169, 17.87641256636620115997628881121, 18.93848836651635313512249730379, 19.46638715118433574189026732750, 20.05673684917198431363307465258, 20.72176959406186343523486861247, 21.75047343802629783924738298655, 22.13373863258863335150093069922

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