Properties

Degree 2
Conductor $ 7 \cdot 11 \cdot 13 $
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 − 4-s − 2·5-s − 7-s + 3·8-s − 3·9-s + 2·10-s + 11-s − 13-s + 14-s − 16-s − 2·17-s + 3·18-s − 4·19-s + 2·20-s − 22-s − 25-s + 26-s + 28-s + 6·29-s − 4·31-s − 5·32-s + 2·34-s + 2·35-s + 3·36-s + 6·37-s + 4·38-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 0.894·5-s − 0.377·7-s + 1.06·8-s − 9-s + 0.632·10-s + 0.301·11-s − 0.277·13-s + 0.267·14-s − 1/4·16-s − 0.485·17-s + 0.707·18-s − 0.917·19-s + 0.447·20-s − 0.213·22-s − 1/5·25-s + 0.196·26-s + 0.188·28-s + 1.11·29-s − 0.718·31-s − 0.883·32-s + 0.342·34-s + 0.338·35-s + 1/2·36-s + 0.986·37-s + 0.648·38-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1001\)    =    \(7 \cdot 11 \cdot 13\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1001} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 1001,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.4784304887$
$L(\frac12)$  $\approx$  $0.4784304887$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{7,\;11,\;13\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{7,\;11,\;13\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad7 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
3 \( 1 + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−19.77870426167325, −19.48896119269777, −18.98093804067506, −18.02705049415991, −17.41446394600451, −16.87689561846359, −16.10001169953917, −15.41812991619613, −14.41371762268618, −14.01536515602809, −12.89352716635308, −12.36295945215812, −11.25398444419577, −10.87450713817732, −9.751006238980820, −9.093925352010554, −8.347715086926500, −7.781155942325061, −6.766134623842887, −5.719160101415377, −4.518719050007294, −3.816301067084477, −2.465107463083540, −0.5806051793737345, 0.5806051793737345, 2.465107463083540, 3.816301067084477, 4.518719050007294, 5.719160101415377, 6.766134623842887, 7.781155942325061, 8.347715086926500, 9.093925352010554, 9.751006238980820, 10.87450713817732, 11.25398444419577, 12.36295945215812, 12.89352716635308, 14.01536515602809, 14.41371762268618, 15.41812991619613, 16.10001169953917, 16.87689561846359, 17.41446394600451, 18.02705049415991, 18.98093804067506, 19.48896119269777, 19.77870426167325

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