Properties

Degree 2
Conductor $ 2 \cdot 5^{2} \cdot 17 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 2·3-s + 4-s − 2·6-s + 4·7-s − 8-s + 9-s + 6·11-s + 2·12-s − 2·13-s − 4·14-s + 16-s + 17-s − 18-s − 4·19-s + 8·21-s − 6·22-s − 2·24-s + 2·26-s − 4·27-s + 4·28-s − 4·31-s − 32-s + 12·33-s − 34-s + 36-s + 4·37-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s + 1.80·11-s + 0.577·12-s − 0.554·13-s − 1.06·14-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.917·19-s + 1.74·21-s − 1.27·22-s − 0.408·24-s + 0.392·26-s − 0.769·27-s + 0.755·28-s − 0.718·31-s − 0.176·32-s + 2.08·33-s − 0.171·34-s + 1/6·36-s + 0.657·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(850\)    =    \(2 \cdot 5^{2} \cdot 17\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{850} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 850,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.010521760\)
\(L(\frac12)\)  \(\approx\)  \(2.010521760\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

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

Imaginary part of the first few zeros on the critical line

−19.73404627341347, −19.39978646463892, −18.47111243602041, −17.73060721634523, −17.04212589996412, −16.60440705714767, −15.21235076793888, −14.67291855405254, −14.46675716280353, −13.59000345950258, −12.38117203649054, −11.60554522575739, −11.03240621996834, −9.918830014197729, −9.069887952032692, −8.627278891970763, −7.861987073989801, −7.131821009443888, −5.995558752633652, −4.615337327117587, −3.671138975930027, −2.303598743324904, −1.444155707931048, 1.444155707931048, 2.303598743324904, 3.671138975930027, 4.615337327117587, 5.995558752633652, 7.131821009443888, 7.861987073989801, 8.627278891970763, 9.069887952032692, 9.918830014197729, 11.03240621996834, 11.60554522575739, 12.38117203649054, 13.59000345950258, 14.46675716280353, 14.67291855405254, 15.21235076793888, 16.60440705714767, 17.04212589996412, 17.73060721634523, 18.47111243602041, 19.39978646463892, 19.73404627341347

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