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Properties

Label 4-5202e2-1.1-c1e2-0-16
Degree $4$
Conductor $27060804$
Sign $1$
Analytic cond. $1725.42$
Root an. cond. $6.44501$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·4-s + 4·8-s − 8·13-s + 5·16-s − 8·19-s − 8·25-s − 16·26-s + 6·32-s − 16·38-s − 8·43-s + 16·47-s − 14·49-s − 16·50-s − 24·52-s − 8·53-s − 8·59-s + 7·64-s − 24·67-s − 24·76-s − 8·83-s − 16·86-s + 32·94-s − 28·98-s − 24·100-s − 24·101-s − 32·103-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 1.41·8-s − 2.21·13-s + 5/4·16-s − 1.83·19-s − 8/5·25-s − 3.13·26-s + 1.06·32-s − 2.59·38-s − 1.21·43-s + 2.33·47-s − 2·49-s − 2.26·50-s − 3.32·52-s − 1.09·53-s − 1.04·59-s + 7/8·64-s − 2.93·67-s − 2.75·76-s − 0.878·83-s − 1.72·86-s + 3.30·94-s − 2.82·98-s − 2.39·100-s − 2.38·101-s − 3.15·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27060804 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27060804 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(27060804\)    =    \(2^{2} \cdot 3^{4} \cdot 17^{4}\)
Sign: $1$
Analytic conductor: \(1725.42\)
Root analytic conductor: \(6.44501\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 27060804,\ (\ :1/2, 1/2),\ 1)\)

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)$Isogeny Class over $\mathbf{F}_p$
bad2$C_1$ \( ( 1 - T )^{2} \)
3 \( 1 \)
17 \( 1 \)
good5$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.5.a_i
7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.11.a_ak
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.13.i_bq
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.23.a_o
29$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \) 2.29.a_bo
31$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.31.a_be
37$C_2^2$ \( 1 + 56 T^{2} + p^{2} T^{4} \) 2.37.a_ce
41$C_2^2$ \( 1 + 80 T^{2} + p^{2} T^{4} \) 2.41.a_dc
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.47.aq_gc
53$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.53.i_es
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.59.i_fe
61$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \) 2.61.a_abo
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.67.y_ks
71$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \) 2.71.a_eg
73$C_2^2$ \( 1 + 96 T^{2} + p^{2} T^{4} \) 2.73.a_ds
79$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.79.a_be
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.83.i_ha
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2^2$ \( 1 + 176 T^{2} + p^{2} T^{4} \) 2.97.a_gu
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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

−7.75380878681612994749927365382, −7.68582147448511375802227195743, −7.08723221152403327686384520603, −7.05619607850836071291234291477, −6.35603801716871027830416335341, −6.30077247036049093190895201608, −5.73201587559646545670012627853, −5.54535162690106992289357885630, −4.91948267407525593483839240781, −4.72029342750362956367587249574, −4.40777553926114437731209101934, −4.02296271875790227656328213350, −3.65382085474867776153686335678, −2.99697442188261949983868665252, −2.64388474953405160966341222365, −2.41615921997312409856063197069, −1.67006343797710253516367950950, −1.58574485714918713104991789401, 0, 0, 1.58574485714918713104991789401, 1.67006343797710253516367950950, 2.41615921997312409856063197069, 2.64388474953405160966341222365, 2.99697442188261949983868665252, 3.65382085474867776153686335678, 4.02296271875790227656328213350, 4.40777553926114437731209101934, 4.72029342750362956367587249574, 4.91948267407525593483839240781, 5.54535162690106992289357885630, 5.73201587559646545670012627853, 6.30077247036049093190895201608, 6.35603801716871027830416335341, 7.05619607850836071291234291477, 7.08723221152403327686384520603, 7.68582147448511375802227195743, 7.75380878681612994749927365382

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