Properties

Label 4-48e4-1.1-c3e2-0-10
Degree $4$
Conductor $5308416$
Sign $1$
Analytic cond. $18479.7$
Root an. cond. $11.6593$
Motivic weight $3$
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  + 180·17-s + 232·19-s − 142·25-s − 108·41-s + 40·43-s − 674·49-s − 648·59-s − 232·67-s − 2.21e3·73-s + 2.30e3·83-s + 1.83e3·89-s + 380·97-s + 504·107-s + 4.42e3·113-s − 2.66e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.32e3·169-s + 173-s + 179-s + ⋯
L(s)  = 1  + 2.56·17-s + 2.80·19-s − 1.13·25-s − 0.411·41-s + 0.141·43-s − 1.96·49-s − 1.42·59-s − 0.423·67-s − 3.54·73-s + 3.04·83-s + 2.18·89-s + 0.397·97-s + 0.455·107-s + 3.68·113-s − 2·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 0.601·169-s + 0.000439·173-s + 0.000417·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5308416\)    =    \(2^{16} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(18479.7\)
Root analytic conductor: \(11.6593\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5308416,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(4.617963143\)
\(L(\frac12)\) \(\approx\) \(4.617963143\)
\(L(\frac{5}{2})\) 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 \( 1 \)
3 \( 1 \)
good5$C_2^2$ \( 1 + 142 T^{2} + p^{6} T^{4} \)
7$C_2^2$ \( 1 + 674 T^{2} + p^{6} T^{4} \)
11$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
13$C_2^2$ \( 1 + 1322 T^{2} + p^{6} T^{4} \)
17$C_2$ \( ( 1 - 90 T + p^{3} T^{2} )^{2} \)
19$C_2$ \( ( 1 - 116 T + p^{3} T^{2} )^{2} \)
23$C_2^2$ \( 1 + 13534 T^{2} + p^{6} T^{4} \)
29$C_2^2$ \( 1 - 18722 T^{2} + p^{6} T^{4} \)
31$C_2^2$ \( 1 - 31246 T^{2} + p^{6} T^{4} \)
37$C_2^2$ \( 1 + 100106 T^{2} + p^{6} T^{4} \)
41$C_2$ \( ( 1 + 54 T + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 - 20 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 51694 T^{2} + p^{6} T^{4} \)
53$C_2^2$ \( 1 + 59182 T^{2} + p^{6} T^{4} \)
59$C_2$ \( ( 1 + 324 T + p^{3} T^{2} )^{2} \)
61$C_2^2$ \( 1 + 123290 T^{2} + p^{6} T^{4} \)
67$C_2$ \( ( 1 + 116 T + p^{3} T^{2} )^{2} \)
71$C_2^2$ \( 1 - 497666 T^{2} + p^{6} T^{4} \)
73$C_2$ \( ( 1 + 1106 T + p^{3} T^{2} )^{2} \)
79$C_2^2$ \( 1 + 963890 T^{2} + p^{6} T^{4} \)
83$C_2$ \( ( 1 - 1152 T + p^{3} T^{2} )^{2} \)
89$C_2$ \( ( 1 - 918 T + p^{3} T^{2} )^{2} \)
97$C_2$ \( ( 1 - 190 T + p^{3} T^{2} )^{2} \)
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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

−8.830744779445334484764803017846, −8.456603692256435693296168971668, −7.80417738634900187956603039888, −7.69928050797399009180060390460, −7.46475177973398482117660649937, −7.17679978819873965798763398078, −6.37009355772922835503151262003, −6.00093313949407255418482834444, −5.78028565178077883088594458696, −5.32791366303549614004120650111, −4.79587067400114867246391413552, −4.74490303475695995076106125433, −3.66666250479855471757686946810, −3.57927132445262858466543213481, −3.02424263065069476837742375124, −2.92925385026552880846325082513, −1.73439316377000511640944400436, −1.63879876728257383287394127954, −0.892707165724018343291398271702, −0.52321336631476145320325708580, 0.52321336631476145320325708580, 0.892707165724018343291398271702, 1.63879876728257383287394127954, 1.73439316377000511640944400436, 2.92925385026552880846325082513, 3.02424263065069476837742375124, 3.57927132445262858466543213481, 3.66666250479855471757686946810, 4.74490303475695995076106125433, 4.79587067400114867246391413552, 5.32791366303549614004120650111, 5.78028565178077883088594458696, 6.00093313949407255418482834444, 6.37009355772922835503151262003, 7.17679978819873965798763398078, 7.46475177973398482117660649937, 7.69928050797399009180060390460, 7.80417738634900187956603039888, 8.456603692256435693296168971668, 8.830744779445334484764803017846

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