Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 1667 $
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 − 2·5-s + 6-s − 2·7-s − 8-s + 9-s + 2·10-s − 12-s − 2·13-s + 2·14-s + 2·15-s + 16-s + 6·17-s − 18-s + 4·19-s − 2·20-s + 2·21-s + 2·23-s + 24-s − 25-s + 2·26-s − 27-s − 2·28-s + 2·29-s − 2·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.288·12-s − 0.554·13-s + 0.534·14-s + 0.516·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.917·19-s − 0.447·20-s + 0.436·21-s + 0.417·23-s + 0.204·24-s − 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.377·28-s + 0.371·29-s − 0.365·30-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 10002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 10002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(10002\)    =    \(2 \cdot 3 \cdot 1667\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{10002} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 10002,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.7521474142$
$L(\frac12)$  $\approx$  $0.7521474142$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;1667\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;1667\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
3 \( 1 + T \)
1667 \( 1 - T \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−16.64942739825903, −16.11092736271900, −15.68091902001925, −15.25251210081259, −14.37038809994195, −13.88448439147342, −12.88449050223939, −12.41647761333513, −11.86182420097089, −11.53851239215092, −10.70344124341372, −10.12376897912923, −9.631131961542838, −9.088329612193720, −8.029179529252612, −7.736476447026688, −7.123346671599011, −6.389634549239950, −5.755685269735279, −4.976712093769295, −4.141042598995585, −3.268184081999317, −2.725053667828908, −1.305573526178041, −0.5237333088786354, 0.5237333088786354, 1.305573526178041, 2.725053667828908, 3.268184081999317, 4.141042598995585, 4.976712093769295, 5.755685269735279, 6.389634549239950, 7.123346671599011, 7.736476447026688, 8.029179529252612, 9.088329612193720, 9.631131961542838, 10.12376897912923, 10.70344124341372, 11.53851239215092, 11.86182420097089, 12.41647761333513, 12.88449050223939, 13.88448439147342, 14.37038809994195, 15.25251210081259, 15.68091902001925, 16.11092736271900, 16.64942739825903

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