Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 167 $
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 + 6-s − 8-s + 9-s − 12-s − 4·13-s + 16-s + 6·17-s − 18-s + 4·23-s + 24-s − 5·25-s + 4·26-s − 27-s + 6·29-s − 32-s − 6·34-s + 36-s − 4·37-s + 4·39-s − 10·41-s + 10·43-s − 4·46-s + 8·47-s − 48-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.288·12-s − 1.10·13-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.834·23-s + 0.204·24-s − 25-s + 0.784·26-s − 0.192·27-s + 1.11·29-s − 0.176·32-s − 1.02·34-s + 1/6·36-s − 0.657·37-s + 0.640·39-s − 1.56·41-s + 1.52·43-s − 0.589·46-s + 1.16·47-s − 0.144·48-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1002\)    =    \(2 \cdot 3 \cdot 167\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1002} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 1002,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8814767831$
$L(\frac12)$  $\approx$  $0.8814767831$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;167\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;167\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
3 \( 1 + T \)
167 \( 1 - T \)
good5 \( 1 + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
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

−19.27226516394268, −18.88277047562878, −18.01156347287310, −17.26314952588669, −16.97084701385612, −16.12625351929762, −15.45404205989591, −14.64710186664026, −13.89973057512237, −12.79555396707818, −12.06350476262368, −11.64289580478975, −10.50968897607609, −10.06559257649550, −9.297128212952393, −8.265244470170909, −7.454143826673161, −6.776875827764956, −5.683324962759835, −4.953714318602154, −3.583139528525280, −2.308402017746368, −0.8393448273245080, 0.8393448273245080, 2.308402017746368, 3.583139528525280, 4.953714318602154, 5.683324962759835, 6.776875827764956, 7.454143826673161, 8.265244470170909, 9.297128212952393, 10.06559257649550, 10.50968897607609, 11.64289580478975, 12.06350476262368, 12.79555396707818, 13.89973057512237, 14.64710186664026, 15.45404205989591, 16.12625351929762, 16.97084701385612, 17.26314952588669, 18.01156347287310, 18.88277047562878, 19.27226516394268

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