Properties

Degree 2
Conductor $ 2 \cdot 3 \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 + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s + 11-s + 12-s − 13-s + 14-s − 15-s + 16-s + 4·17-s + 18-s + 6·19-s − 20-s + 21-s + 22-s + 3·23-s + 24-s − 4·25-s − 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 + 0.301·11-s + 0.288·12-s − 0.277·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.970·17-s + 0.235·18-s + 1.37·19-s − 0.223·20-s + 0.218·21-s + 0.213·22-s + 0.625·23-s + 0.204·24-s − 4/5·25-s − 0.196·26-s + 0.192·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 858 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 858 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(858\)    =    \(2 \cdot 3 \cdot 11 \cdot 13\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{858} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 858,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.952932622$
$L(\frac12)$  $\approx$  $2.952932622$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;11,\;13\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11,\;13\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
3 \( 1 - T \)
11 \( 1 - T \)
13 \( 1 + T \)
good5 \( 1 + T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
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

−19.92218263676901, −19.30352746451219, −18.56969676036204, −17.75313495405540, −16.74043338424189, −16.19743026397099, −15.25732252592226, −14.80346532605416, −14.09977307168442, −13.40758547048187, −12.62980867439538, −11.66015642318120, −11.39652627218295, −10.03832110735203, −9.484044853046120, −8.200684149509430, −7.680221455050012, −6.807550865772327, −5.615155656920649, −4.754241436265973, −3.705135195878283, −2.919215070324715, −1.455293974952882, 1.455293974952882, 2.919215070324715, 3.705135195878283, 4.754241436265973, 5.615155656920649, 6.807550865772327, 7.680221455050012, 8.200684149509430, 9.484044853046120, 10.03832110735203, 11.39652627218295, 11.66015642318120, 12.62980867439538, 13.40758547048187, 14.09977307168442, 14.80346532605416, 15.25732252592226, 16.19743026397099, 16.74043338424189, 17.75313495405540, 18.56969676036204, 19.30352746451219, 19.92218263676901

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