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Properties

Label 4-34e4-1.1-c0e2-0-2
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 4·13-s + 16-s − 4·52-s − 64-s − 81-s + 4·89-s − 4·101-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 4·208-s + 211-s + ⋯
L(s)  = 1  − 4-s + 4·13-s + 16-s − 4·52-s − 64-s − 81-s + 4·89-s − 4·101-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 4·208-s + 211-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

\(L(\frac{1}{2})\) \(\approx\) \(1.021881552\)
\(L(\frac12)\) \(\approx\) \(1.021881552\)
\(L(1)\) 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$C_2^2$ \( 1 + T^{4} \)
7$C_2^2$ \( 1 + T^{4} \)
11$C_2^2$ \( 1 + T^{4} \)
13$C_1$ \( ( 1 - T )^{4} \)
19$C_2$ \( ( 1 + T^{2} )^{2} \)
23$C_2^2$ \( 1 + T^{4} \)
29$C_2^2$ \( 1 + T^{4} \)
31$C_2^2$ \( 1 + T^{4} \)
37$C_2^2$ \( 1 + T^{4} \)
41$C_2^2$ \( 1 + T^{4} \)
43$C_2$ \( ( 1 + T^{2} )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_2$ \( ( 1 + T^{2} )^{2} \)
59$C_2$ \( ( 1 + T^{2} )^{2} \)
61$C_2^2$ \( 1 + T^{4} \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
71$C_2^2$ \( 1 + T^{4} \)
73$C_2^2$ \( 1 + T^{4} \)
79$C_2^2$ \( 1 + T^{4} \)
83$C_2$ \( ( 1 + T^{2} )^{2} \)
89$C_1$ \( ( 1 - T )^{4} \)
97$C_2^2$ \( 1 + T^{4} \)
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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

−10.14911886812605344966685252029, −9.828436684303266026564496717583, −9.042770923072000633582516740499, −9.042761878918763567807525692038, −8.528243929486229495731371508990, −8.443073143215756793929554480558, −7.73351118326242519059924752766, −7.61291123898062320768263169986, −6.60075638751644783424001245416, −6.34509077012111079513952371251, −6.12474227824019852974080244415, −5.48307811789978765772994242419, −5.21657532887193524305873784777, −4.46382693193862481546202965180, −3.96952724505042743645441889020, −3.63349675825252596942810395588, −3.39118794709560083732820756058, −2.50896669731885138429607978795, −1.34928060120166883895863755008, −1.20800603005854348550064001621, 1.20800603005854348550064001621, 1.34928060120166883895863755008, 2.50896669731885138429607978795, 3.39118794709560083732820756058, 3.63349675825252596942810395588, 3.96952724505042743645441889020, 4.46382693193862481546202965180, 5.21657532887193524305873784777, 5.48307811789978765772994242419, 6.12474227824019852974080244415, 6.34509077012111079513952371251, 6.60075638751644783424001245416, 7.61291123898062320768263169986, 7.73351118326242519059924752766, 8.443073143215756793929554480558, 8.528243929486229495731371508990, 9.042761878918763567807525692038, 9.042770923072000633582516740499, 9.828436684303266026564496717583, 10.14911886812605344966685252029

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