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Properties

Label 4-34e4-1.1-c0e2-0-1
Degree $4$
Conductor $1336336$
Sign $1$
Analytic cond. $0.332835$
Root an. cond. $0.759551$
Motivic weight $0$
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  − 4-s + 2·5-s + 16-s − 2·20-s + 2·25-s − 2·29-s + 2·37-s − 2·41-s + 2·61-s − 64-s − 2·73-s + 2·80-s − 81-s + 2·97-s − 2·100-s + 2·109-s + 2·113-s + 2·116-s + 2·125-s + 127-s + 131-s + 137-s + 139-s − 4·145-s − 2·148-s + 149-s + 151-s + ⋯
L(s)  = 1  − 4-s + 2·5-s + 16-s − 2·20-s + 2·25-s − 2·29-s + 2·37-s − 2·41-s + 2·61-s − 64-s − 2·73-s + 2·80-s − 81-s + 2·97-s − 2·100-s + 2·109-s + 2·113-s + 2·116-s + 2·125-s + 127-s + 131-s + 137-s + 139-s − 4·145-s − 2·148-s + 149-s + 151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1336336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1336336\)    =    \(2^{4} \cdot 17^{4}\)
Sign: $1$
Analytic conductor: \(0.332835\)
Root analytic conductor: \(0.759551\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1336336,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.170997959\)
\(L(\frac12)\) \(\approx\) \(1.170997959\)
\(L(1)\) 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)$
bad2$C_2$ \( 1 + T^{2} \)
17 \( 1 \)
good3$C_2^2$ \( 1 + T^{4} \)
5$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
7$C_2^2$ \( 1 + T^{4} \)
11$C_2^2$ \( 1 + T^{4} \)
13$C_2$ \( ( 1 + T^{2} )^{2} \)
19$C_2$ \( ( 1 + T^{2} )^{2} \)
23$C_2^2$ \( 1 + T^{4} \)
29$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
31$C_2^2$ \( 1 + T^{4} \)
37$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
41$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
43$C_2$ \( ( 1 + T^{2} )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
59$C_2$ \( ( 1 + T^{2} )^{2} \)
61$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
71$C_2^2$ \( 1 + T^{4} \)
73$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
79$C_2^2$ \( 1 + T^{4} \)
83$C_2$ \( ( 1 + T^{2} )^{2} \)
89$C_2$ \( ( 1 + T^{2} )^{2} \)
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

−10.00970584902981600579973287175, −9.652845597817256245731247013466, −9.607415386083401094956271544170, −9.062988826822614822111352980273, −8.595052282691261720801964921958, −8.439657804706711083837858769858, −7.72879367723669114854597092930, −7.29810120254522126109350699028, −6.87669512574765925648394428030, −6.16740746409478676644620606513, −5.97990528087817090924620174529, −5.42975100836414362453355487544, −5.37946989256413027767597702369, −4.50164390692390560834591818762, −4.38084617078799069393105239350, −3.41317344429636947129541897105, −3.18467872154381935110746738526, −2.13342109937984989800887388856, −1.97516299636307899317861391759, −1.06916387095544160553343369755, 1.06916387095544160553343369755, 1.97516299636307899317861391759, 2.13342109937984989800887388856, 3.18467872154381935110746738526, 3.41317344429636947129541897105, 4.38084617078799069393105239350, 4.50164390692390560834591818762, 5.37946989256413027767597702369, 5.42975100836414362453355487544, 5.97990528087817090924620174529, 6.16740746409478676644620606513, 6.87669512574765925648394428030, 7.29810120254522126109350699028, 7.72879367723669114854597092930, 8.439657804706711083837858769858, 8.595052282691261720801964921958, 9.062988826822614822111352980273, 9.607415386083401094956271544170, 9.652845597817256245731247013466, 10.00970584902981600579973287175

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