Properties

Label 4-39e4-1.1-c1e2-0-3
Degree $4$
Conductor $2313441$
Sign $1$
Analytic cond. $147.507$
Root an. cond. $3.48500$
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-s + 4-s + 3·5-s − 3·7-s − 3·8-s − 3·10-s + 4·11-s + 3·14-s + 16-s − 17-s + 6·19-s + 3·20-s − 4·22-s − 4·23-s + 25-s − 3·28-s − 29-s + 31-s + 32-s + 34-s − 9·35-s + 11·37-s − 6·38-s − 9·40-s − 41-s + 5·43-s + 4·44-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 1.34·5-s − 1.13·7-s − 1.06·8-s − 0.948·10-s + 1.20·11-s + 0.801·14-s + 1/4·16-s − 0.242·17-s + 1.37·19-s + 0.670·20-s − 0.852·22-s − 0.834·23-s + 1/5·25-s − 0.566·28-s − 0.185·29-s + 0.179·31-s + 0.176·32-s + 0.171·34-s − 1.52·35-s + 1.80·37-s − 0.973·38-s − 1.42·40-s − 0.156·41-s + 0.762·43-s + 0.603·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2313441\)    =    \(3^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(147.507\)
Root analytic conductor: \(3.48500\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1521} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2313441,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.602008771\)
\(L(\frac12)\) \(\approx\) \(1.602008771\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
13 \( 1 \)
good2$D_{4}$ \( 1 + T + p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 3 T + 12 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 + T + 30 T^{2} + p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 + T + 20 T^{2} + p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - T + 58 T^{2} - p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 11 T + 100 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + T + 78 T^{2} + p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 5 T + 88 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 11 T + 98 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 14 T + 150 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 16 T + 169 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 5 T + 136 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
73$D_{4}$ \( 1 + 12 T + 165 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 15 T + 210 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 10 T + 174 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 18 T + 242 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 13 T + 232 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
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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

−9.575064592006727561361200432287, −9.514364039626708053858943058344, −8.932062938579776362028256459747, −8.761932484082927400943705391090, −8.096298138108797264218175846685, −7.74163278275719554611187347649, −7.13424607697466870171199888313, −6.65915589160477294584562458524, −6.58175051174562119400926562059, −5.89205931019696178410209558765, −5.87558146837307924857942616510, −5.44924401206447988889956578646, −4.56840936788280657444995943021, −4.16497594149603013926107489926, −3.49139938350712492155446945662, −3.06888486893489483481767294671, −2.55006818817911547437275004447, −2.02109197799381099253432116778, −1.36103810056341249886701136358, −0.57968049168876909252866929188, 0.57968049168876909252866929188, 1.36103810056341249886701136358, 2.02109197799381099253432116778, 2.55006818817911547437275004447, 3.06888486893489483481767294671, 3.49139938350712492155446945662, 4.16497594149603013926107489926, 4.56840936788280657444995943021, 5.44924401206447988889956578646, 5.87558146837307924857942616510, 5.89205931019696178410209558765, 6.58175051174562119400926562059, 6.65915589160477294584562458524, 7.13424607697466870171199888313, 7.74163278275719554611187347649, 8.096298138108797264218175846685, 8.761932484082927400943705391090, 8.932062938579776362028256459747, 9.514364039626708053858943058344, 9.575064592006727561361200432287

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