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Properties

Label 4-1700e2-1.1-c0e2-0-1
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  − 2·2-s + 3·4-s − 4·8-s + 5·16-s − 2·29-s − 6·32-s + 2·37-s + 2·41-s + 4·58-s − 2·61-s + 7·64-s + 2·73-s − 4·74-s − 81-s − 4·82-s + 2·97-s + 2·109-s + 2·113-s − 6·116-s + 4·122-s + 127-s − 8·128-s + 131-s + 137-s + 139-s − 4·146-s + 6·148-s + ⋯
L(s)  = 1  − 2·2-s + 3·4-s − 4·8-s + 5·16-s − 2·29-s − 6·32-s + 2·37-s + 2·41-s + 4·58-s − 2·61-s + 7·64-s + 2·73-s − 4·74-s − 81-s − 4·82-s + 2·97-s + 2·109-s + 2·113-s − 6·116-s + 4·122-s + 127-s − 8·128-s + 131-s + 137-s + 139-s − 4·146-s + 6·148-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

\(L(\frac{1}{2})\) \(\approx\) \(0.4439077579\)
\(L(\frac12)\) \(\approx\) \(0.4439077579\)
\(L(1)\) 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$C_1$ \( ( 1 + T )^{2} \)
5 \( 1 \)
17$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 + T^{4} \)
7$C_2^2$ \( 1 + T^{4} \)
11$C_2^2$ \( 1 + T^{4} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_2$ \( ( 1 + T^{2} )^{2} \)
53$C_2$ \( ( 1 + T^{2} )^{2} \)
59$C_2$ \( ( 1 + T^{2} )^{2} \)
61$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
67$C_2$ \( ( 1 + T^{2} )^{2} \)
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_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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

−9.675973171610741880002485233631, −9.316078711510165378088038248835, −8.864938947533954012293387069381, −8.839197949957727038205227031445, −8.079531703049957269561222649299, −7.70770368266227088634182141654, −7.57743247305971626789641472862, −7.27671312364989155538289425095, −6.61146551232372573813698466420, −6.26159094303157856638113848005, −5.82836971516721057631842478349, −5.67228740783405640674547744168, −4.84714458698685385258314122428, −4.26212232724169480419943684653, −3.57656010934202825583265471919, −3.20387380085439095013493761912, −2.45180026866255025283259880845, −2.19685911830970152693719719983, −1.47283931014291734955803349509, −0.73529168076450384789291264501, 0.73529168076450384789291264501, 1.47283931014291734955803349509, 2.19685911830970152693719719983, 2.45180026866255025283259880845, 3.20387380085439095013493761912, 3.57656010934202825583265471919, 4.26212232724169480419943684653, 4.84714458698685385258314122428, 5.67228740783405640674547744168, 5.82836971516721057631842478349, 6.26159094303157856638113848005, 6.61146551232372573813698466420, 7.27671312364989155538289425095, 7.57743247305971626789641472862, 7.70770368266227088634182141654, 8.079531703049957269561222649299, 8.839197949957727038205227031445, 8.864938947533954012293387069381, 9.316078711510165378088038248835, 9.675973171610741880002485233631

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