Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 131 $
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 − 8-s + 9-s − 2·10-s + 4·11-s − 12-s − 2·13-s − 2·15-s + 16-s − 2·17-s − 18-s + 8·19-s + 2·20-s − 4·22-s + 24-s − 25-s + 2·26-s − 27-s + 2·29-s + 2·30-s − 8·31-s − 32-s − 4·33-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 1.20·11-s − 0.288·12-s − 0.554·13-s − 0.516·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 1.83·19-s + 0.447·20-s − 0.852·22-s + 0.204·24-s − 1/5·25-s + 0.392·26-s − 0.192·27-s + 0.371·29-s + 0.365·30-s − 1.43·31-s − 0.176·32-s − 0.696·33-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 786 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 786 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(786\)    =    \(2 \cdot 3 \cdot 131\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{786} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 786,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.149014222$
$L(\frac12)$  $\approx$  $1.149014222$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;131\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;131\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
131 \( 1 - T \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−19.89286166564976, −19.35024993031709, −18.29647247362728, −17.80545310019151, −17.36722536054577, −16.50676684771621, −16.09315882168020, −14.97102927796713, −14.24587286886627, −13.47420493909041, −12.48172632441282, −11.70780418651535, −11.12579620129703, −10.04730012038988, −9.535725038220236, −8.901349969419321, −7.570487819496096, −6.872819579434118, −5.961706122456092, −5.200104365341331, −3.800368480473856, −2.288266018943178, −1.074236921916057, 1.074236921916057, 2.288266018943178, 3.800368480473856, 5.200104365341331, 5.961706122456092, 6.872819579434118, 7.570487819496096, 8.901349969419321, 9.535725038220236, 10.04730012038988, 11.12579620129703, 11.70780418651535, 12.48172632441282, 13.47420493909041, 14.24587286886627, 14.97102927796713, 16.09315882168020, 16.50676684771621, 17.36722536054577, 17.80545310019151, 18.29647247362728, 19.35024993031709, 19.89286166564976

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