Properties

Label 4-1700e2-1.1-c0e2-0-3
Degree $4$
Conductor $2890000$
Sign $1$
Analytic cond. $0.719800$
Root an. cond. $0.921092$
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 + 16-s + 2·17-s + 2·29-s + 2·37-s + 2·41-s − 2·61-s − 64-s − 2·68-s − 2·73-s − 81-s + 2·97-s − 2·109-s + 2·113-s − 2·116-s + 127-s + 131-s + 137-s + 139-s − 2·148-s + 149-s + 151-s + 157-s + 163-s − 2·164-s + 167-s − 2·169-s + ⋯
L(s)  = 1  − 4-s + 16-s + 2·17-s + 2·29-s + 2·37-s + 2·41-s − 2·61-s − 64-s − 2·68-s − 2·73-s − 81-s + 2·97-s − 2·109-s + 2·113-s − 2·116-s + 127-s + 131-s + 137-s + 139-s − 2·148-s + 149-s + 151-s + 157-s + 163-s − 2·164-s + 167-s − 2·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2890000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2890000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2890000\)    =    \(2^{4} \cdot 5^{4} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(0.719800\)
Root analytic conductor: \(0.921092\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2890000,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.058398888\)
\(L(\frac12)\) \(\approx\) \(1.058398888\)
\(L(1)\) 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$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
17$C_1$ \( ( 1 - T )^{2} \)
good3$C_2^2$ \( 1 + T^{4} \)
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

−9.621693204528338566444015964958, −9.551540258418744755566180179705, −8.745725562750514140642658078143, −8.739553007836925901026062919836, −8.049062540152930126437150533153, −7.904385708032775234005493040217, −7.29929746361582017316779981568, −7.29311814974168229559373964873, −6.22981557822582367690271098918, −5.97799437245419381017560834066, −5.90844963570508377150311121060, −5.13609979479622221374425467642, −4.72839447478391209968383154611, −4.42529315141504950159012268127, −3.94661804210477456945265977563, −3.32032866383993487822211509574, −2.93263388386914010013786930941, −2.46015059581200257226223753034, −1.28997041010415709024147484752, −0.987411808212863579655714939590, 0.987411808212863579655714939590, 1.28997041010415709024147484752, 2.46015059581200257226223753034, 2.93263388386914010013786930941, 3.32032866383993487822211509574, 3.94661804210477456945265977563, 4.42529315141504950159012268127, 4.72839447478391209968383154611, 5.13609979479622221374425467642, 5.90844963570508377150311121060, 5.97799437245419381017560834066, 6.22981557822582367690271098918, 7.29311814974168229559373964873, 7.29929746361582017316779981568, 7.904385708032775234005493040217, 8.049062540152930126437150533153, 8.739553007836925901026062919836, 8.745725562750514140642658078143, 9.551540258418744755566180179705, 9.621693204528338566444015964958

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