Properties

Label 4-448e2-1.1-c6e2-0-1
Degree $4$
Conductor $200704$
Sign $1$
Analytic cond. $10622.2$
Root an. cond. $10.1520$
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  − 266·7-s − 582·9-s + 1.74e3·11-s − 9.47e3·23-s + 2.92e4·25-s − 2.22e4·29-s − 6.00e3·37-s + 6.28e4·43-s − 4.68e4·49-s + 1.52e5·53-s + 1.54e5·63-s + 9.90e5·67-s + 3.68e5·71-s − 4.64e5·77-s + 1.06e6·79-s − 1.92e5·81-s − 1.01e6·99-s + 3.23e6·107-s − 3.98e5·109-s − 3.61e6·113-s − 1.25e6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  − 0.775·7-s − 0.798·9-s + 1.31·11-s − 0.778·23-s + 1.86·25-s − 0.914·29-s − 0.118·37-s + 0.790·43-s − 0.398·49-s + 1.02·53-s + 0.619·63-s + 3.29·67-s + 1.03·71-s − 1.01·77-s + 2.16·79-s − 0.362·81-s − 1.04·99-s + 2.63·107-s − 0.307·109-s − 2.50·113-s − 0.706·121-s + 0.603·161-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(200704\)    =    \(2^{12} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(10622.2\)
Root analytic conductor: \(10.1520\)
Motivic weight: \(6\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 200704,\ (\ :3, 3),\ 1)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(3.112795454\)
\(L(\frac12)\) \(\approx\) \(3.112795454\)
\(L(4)\) 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 \( 1 \)
7$C_2$ \( 1 + 38 p T + p^{6} T^{2} \)
good3$C_2^2$ \( 1 + 194 p T^{2} + p^{12} T^{4} \)
5$C_2^2$ \( 1 - 5842 p T^{2} + p^{12} T^{4} \)
11$C_2$ \( ( 1 - 874 T + p^{6} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 4755578 T^{2} + p^{12} T^{4} \)
17$C_2^2$ \( 1 - 748834 p T^{2} + p^{12} T^{4} \)
19$C_2^2$ \( 1 - 84379322 T^{2} + p^{12} T^{4} \)
23$C_2$ \( ( 1 + 206 p T + p^{6} T^{2} )^{2} \)
29$C_2$ \( ( 1 + 11146 T + p^{6} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 1020892802 T^{2} + p^{12} T^{4} \)
37$C_2$ \( ( 1 + 3002 T + p^{6} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 6189133442 T^{2} + p^{12} T^{4} \)
43$C_2$ \( ( 1 - 31418 T + p^{6} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 16309886018 T^{2} + p^{12} T^{4} \)
53$C_2$ \( ( 1 - 76406 T + p^{6} T^{2} )^{2} \)
59$C_2^2$ \( 1 - 71539567322 T^{2} + p^{12} T^{4} \)
61$C_2^2$ \( 1 - 27356175482 T^{2} + p^{12} T^{4} \)
67$C_2$ \( ( 1 - 495242 T + p^{6} T^{2} )^{2} \)
71$C_2$ \( ( 1 - 184406 T + p^{6} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 298950552578 T^{2} + p^{12} T^{4} \)
79$C_2$ \( ( 1 - 534934 T + p^{6} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 142873131578 T^{2} + p^{12} T^{4} \)
89$C_2^2$ \( 1 - 597656180162 T^{2} + p^{12} T^{4} \)
97$C_2^2$ \( 1 - 1002631840898 T^{2} + p^{12} T^{4} \)
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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

−10.35975311234755960582448205832, −9.683069924513947417949015360203, −9.507041806854325750020528282384, −8.915033825210525443897666670966, −8.752542796544580267375474777308, −8.037963393590195261749465834626, −7.70249330048348284377377635346, −6.84029259910083236465689432262, −6.62348364268915526646110935035, −6.36869929724235442046223231743, −5.44375800127112965354859244971, −5.39629878087637645162000216612, −4.52127174721434531264762267451, −3.84012117886480853541166187784, −3.61103742007419907209339281895, −2.90547625683293012543094425557, −2.34868696604107579943828905698, −1.68589528426697196652929301677, −0.813408362395890664800792255971, −0.50505755600262888008954180074, 0.50505755600262888008954180074, 0.813408362395890664800792255971, 1.68589528426697196652929301677, 2.34868696604107579943828905698, 2.90547625683293012543094425557, 3.61103742007419907209339281895, 3.84012117886480853541166187784, 4.52127174721434531264762267451, 5.39629878087637645162000216612, 5.44375800127112965354859244971, 6.36869929724235442046223231743, 6.62348364268915526646110935035, 6.84029259910083236465689432262, 7.70249330048348284377377635346, 8.037963393590195261749465834626, 8.752542796544580267375474777308, 8.915033825210525443897666670966, 9.507041806854325750020528282384, 9.683069924513947417949015360203, 10.35975311234755960582448205832

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