Properties

Label 4-39e4-1.1-c0e2-0-1
Degree $4$
Conductor $2313441$
Sign $1$
Analytic cond. $0.576199$
Root an. cond. $0.871250$
Motivic weight $0$
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  + 2·7-s − 16-s + 2·19-s − 2·31-s − 2·37-s + 2·49-s + 2·67-s + 2·73-s − 2·97-s + 2·109-s − 2·112-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  + 2·7-s − 16-s + 2·19-s − 2·31-s − 2·37-s + 2·49-s + 2·67-s + 2·73-s − 2·97-s + 2·109-s − 2·112-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(2313441\)    =    \(3^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(0.576199\)
Root analytic conductor: \(0.871250\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1521} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2313441,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.444585619\)
\(L(\frac12)\) \(\approx\) \(1.444585619\)
\(L(1)\) 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$C_2^2$ \( 1 + T^{4} \)
5$C_2^2$ \( 1 + T^{4} \)
7$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
11$C_2^2$ \( 1 + T^{4} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
19$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
29$C_2$ \( ( 1 + T^{2} )^{2} \)
31$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
37$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
41$C_2^2$ \( 1 + T^{4} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_2^2$ \( 1 + T^{4} \)
53$C_2$ \( ( 1 + T^{2} )^{2} \)
59$C_2^2$ \( 1 + T^{4} \)
61$C_2$ \( ( 1 + T^{2} )^{2} \)
67$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
71$C_2^2$ \( 1 + T^{4} \)
73$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
79$C_2$ \( ( 1 + T^{2} )^{2} \)
83$C_2^2$ \( 1 + T^{4} \)
89$C_2^2$ \( 1 + T^{4} \)
97$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
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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

−9.771287775595117019428877122743, −9.492389179499621192347036867256, −8.936301586420276716704119228963, −8.680573982591645169164372269874, −8.292715549792970586562583242963, −7.84138029764644006451824924470, −7.46548761150637029561010310272, −7.13800899756430339627551879203, −6.84212553812636642622348832502, −6.17205468239825203767644146189, −5.44778760234932995012295545444, −5.31575483310630235484014456841, −4.98999663676339519289227836343, −4.58013366071585867010715589083, −3.77475559309054979235419571932, −3.64893983664295827796460757339, −2.85421897115998887987872250432, −2.05742189762647548783545866750, −1.81695550304147326141005723360, −1.07304974082192346023987081374, 1.07304974082192346023987081374, 1.81695550304147326141005723360, 2.05742189762647548783545866750, 2.85421897115998887987872250432, 3.64893983664295827796460757339, 3.77475559309054979235419571932, 4.58013366071585867010715589083, 4.98999663676339519289227836343, 5.31575483310630235484014456841, 5.44778760234932995012295545444, 6.17205468239825203767644146189, 6.84212553812636642622348832502, 7.13800899756430339627551879203, 7.46548761150637029561010310272, 7.84138029764644006451824924470, 8.292715549792970586562583242963, 8.680573982591645169164372269874, 8.936301586420276716704119228963, 9.492389179499621192347036867256, 9.771287775595117019428877122743

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