Properties

Label 4-460e2-1.1-c1e2-0-0
Degree $4$
Conductor $211600$
Sign $1$
Analytic cond. $13.4917$
Root an. cond. $1.91653$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·4-s − 2·5-s − 6·9-s − 4·10-s − 4·13-s − 4·16-s + 6·17-s − 12·18-s − 4·20-s + 3·25-s − 8·26-s + 14·29-s − 8·32-s + 12·34-s − 12·36-s + 22·37-s + 2·41-s + 12·45-s − 13·49-s + 6·50-s − 8·52-s + 22·53-s + 28·58-s − 16·61-s − 8·64-s + 8·65-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s − 0.894·5-s − 2·9-s − 1.26·10-s − 1.10·13-s − 16-s + 1.45·17-s − 2.82·18-s − 0.894·20-s + 3/5·25-s − 1.56·26-s + 2.59·29-s − 1.41·32-s + 2.05·34-s − 2·36-s + 3.61·37-s + 0.312·41-s + 1.78·45-s − 1.85·49-s + 0.848·50-s − 1.10·52-s + 3.02·53-s + 3.67·58-s − 2.04·61-s − 64-s + 0.992·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 211600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(211600\)    =    \(2^{4} \cdot 5^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(13.4917\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{211600} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 211600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.153218092\)
\(L(\frac12)\) \(\approx\) \(2.153218092\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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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

−9.254856050792654313639844453491, −8.261450062978304340130490972772, −7.991219017033404836699134811877, −7.88313741440253118524665483256, −6.89664930841244341712826498598, −6.51829449666043836359577697870, −6.00165457583941898386124574131, −5.37273982184814550156636643722, −5.23992404060255109399587286374, −4.32075270011969972226267491278, −4.22484382518970536541722955137, −3.14448167474562522721276000501, −2.87311941525735848219246932072, −2.55324698595565232433692561243, −0.72059586748507280140651813487, 0.72059586748507280140651813487, 2.55324698595565232433692561243, 2.87311941525735848219246932072, 3.14448167474562522721276000501, 4.22484382518970536541722955137, 4.32075270011969972226267491278, 5.23992404060255109399587286374, 5.37273982184814550156636643722, 6.00165457583941898386124574131, 6.51829449666043836359577697870, 6.89664930841244341712826498598, 7.88313741440253118524665483256, 7.991219017033404836699134811877, 8.261450062978304340130490972772, 9.254856050792654313639844453491

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