Properties

Degree 4
Conductor $ 2^{2} \cdot 5^{2} \cdot 17^{3} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 1

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s + 4-s − 8·7-s + 6·9-s − 4·12-s + 16-s − 17-s − 8·19-s + 32·21-s − 5·25-s + 4·27-s − 8·28-s + 6·36-s − 8·37-s − 4·48-s + 34·49-s + 4·51-s + 32·57-s − 48·63-s + 64-s − 68-s + 4·73-s + 20·75-s − 8·76-s − 37·81-s + 32·84-s − 12·89-s + ⋯
L(s)  = 1  − 2.30·3-s + 1/2·4-s − 3.02·7-s + 2·9-s − 1.15·12-s + 1/4·16-s − 0.242·17-s − 1.83·19-s + 6.98·21-s − 25-s + 0.769·27-s − 1.51·28-s + 36-s − 1.31·37-s − 0.577·48-s + 34/7·49-s + 0.560·51-s + 4.23·57-s − 6.04·63-s + 1/8·64-s − 0.121·68-s + 0.468·73-s + 2.30·75-s − 0.917·76-s − 4.11·81-s + 3.49·84-s − 1.27·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 491300 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 491300 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(491300\)    =    \(2^{2} \cdot 5^{2} \cdot 17^{3}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{491300} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((4,\ 491300,\ (\ :1/2, 1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;17\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;5,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_2$ \( 1 + p T^{2} \)
17$C_1$ \( 1 + T \)
good3$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
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\[\begin{aligned}L(s) = \prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.434721631247958880519323524946, −7.54625870597452603613853398245, −6.97765460885046698808229959612, −6.65330731261113455462227374582, −6.39126314971633693713494192915, −5.99911855171913780844071238816, −5.88165569807091607176477515994, −5.21315012084683261869529031894, −4.61516560042655420117257789936, −3.90229547122613769308947697962, −3.39756711777290923826252356515, −2.81090884111852531330067525726, −2.03988638187729415468870477153, −0.56239664780041394316008252541, 0, 0.56239664780041394316008252541, 2.03988638187729415468870477153, 2.81090884111852531330067525726, 3.39756711777290923826252356515, 3.90229547122613769308947697962, 4.61516560042655420117257789936, 5.21315012084683261869529031894, 5.88165569807091607176477515994, 5.99911855171913780844071238816, 6.39126314971633693713494192915, 6.65330731261113455462227374582, 6.97765460885046698808229959612, 7.54625870597452603613853398245, 8.434721631247958880519323524946

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