Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 \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 − 3-s − 4-s + 5-s + 6-s + 7-s + 3·8-s + 9-s − 10-s + 4·11-s + 12-s + 2·13-s − 14-s − 15-s − 16-s + 17-s − 18-s + 8·19-s − 20-s − 21-s − 4·22-s − 4·23-s − 3·24-s + 25-s − 2·26-s − 27-s − 28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s + 1.06·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s + 0.288·12-s + 0.554·13-s − 0.267·14-s − 0.258·15-s − 1/4·16-s + 0.242·17-s − 0.235·18-s + 1.83·19-s − 0.223·20-s − 0.218·21-s − 0.852·22-s − 0.834·23-s − 0.612·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1785 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1785 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1785\)    =    \(3 \cdot 5 \cdot 7 \cdot 17\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1785} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 1785,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.115916506$
$L(\frac12)$  $\approx$  $1.115916506$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;7,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 - T \)
17 \( 1 - T \)
good2 \( 1 + T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−19.56802721460586, −18.67453955418556, −18.21426553859557, −17.71420745508557, −17.22647275487227, −16.31377922033730, −16.20670076681607, −14.89099016805561, −14.19910265123577, −13.69812536800613, −12.98043465704184, −11.91476873213558, −11.56917957366610, −10.54791513980171, −10.00726589282177, −9.213215334459086, −8.769434477106700, −7.693947561099437, −7.097589725743312, −5.974728136161237, −5.348660386455038, −4.360053355875269, −3.510834855358475, −1.721704754074031, −0.9184978877998599, 0.9184978877998599, 1.721704754074031, 3.510834855358475, 4.360053355875269, 5.348660386455038, 5.974728136161237, 7.097589725743312, 7.693947561099437, 8.769434477106700, 9.213215334459086, 10.00726589282177, 10.54791513980171, 11.56917957366610, 11.91476873213558, 12.98043465704184, 13.69812536800613, 14.19910265123577, 14.89099016805561, 16.20670076681607, 16.31377922033730, 17.22647275487227, 17.71420745508557, 18.21426553859557, 18.67453955418556, 19.56802721460586

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