Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 167 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 5-s − 6-s − 7-s − 8-s + 9-s + 10-s + 2·11-s + 12-s − 6·13-s + 14-s − 15-s + 16-s − 4·17-s − 18-s − 4·19-s − 20-s − 21-s − 2·22-s + 8·23-s − 24-s − 4·25-s + 6·26-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 − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.603·11-s + 0.288·12-s − 1.66·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.970·17-s − 0.235·18-s − 0.917·19-s − 0.223·20-s − 0.218·21-s − 0.426·22-s + 1.66·23-s − 0.204·24-s − 4/5·25-s + 1.17·26-s + 0.192·27-s − 0.188·28-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  =  1
Selberg data  =  $(2,\ 1002,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;167\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;167\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - T \)
167 \( 1 - T \)
good5 \( 1 + T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 7 T + p T^{2} \)
59 \( 1 - 11 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 15 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−19.60656315764613, −19.34227765911372, −18.67402435070859, −17.66052048005228, −17.05957468909470, −16.51911386107865, −15.51790426835813, −14.95613352029150, −14.51324279050270, −13.26025393996206, −12.72635114730873, −11.79581428958183, −11.12218268705133, −10.15208271613289, −9.424406294099138, −8.851961090123594, −7.964926657257996, −7.094229043592795, −6.603517225165509, −5.115159755814799, −4.064923074300365, −2.937809295599531, −1.899862266331342, 0, 1.899862266331342, 2.937809295599531, 4.064923074300365, 5.115159755814799, 6.603517225165509, 7.094229043592795, 7.964926657257996, 8.851961090123594, 9.424406294099138, 10.15208271613289, 11.12218268705133, 11.79581428958183, 12.72635114730873, 13.26025393996206, 14.51324279050270, 14.95613352029150, 15.51790426835813, 16.51911386107865, 17.05957468909470, 17.66052048005228, 18.67402435070859, 19.34227765911372, 19.60656315764613

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