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 + 4·5-s − 6-s − 4·7-s + 8-s + 9-s + 4·10-s − 12-s + 6·13-s − 4·14-s − 4·15-s + 16-s − 2·17-s + 18-s + 4·19-s + 4·20-s + 4·21-s + 4·23-s − 24-s + 11·25-s + 6·26-s − 27-s − 4·28-s − 6·29-s − 4·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.78·5-s − 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s + 1.26·10-s − 0.288·12-s + 1.66·13-s − 1.06·14-s − 1.03·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.917·19-s + 0.894·20-s + 0.872·21-s + 0.834·23-s − 0.204·24-s + 11/5·25-s + 1.17·26-s − 0.192·27-s − 0.755·28-s − 1.11·29-s − 0.730·30-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  =  $(2,\ 786,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.467708208$
$L(\frac12)$  $\approx$  $2.467708208$
$L(\frac{3}{2})$   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 \( 1 - 4 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−19.25175833806691, −18.48056168104660, −17.85556944710945, −17.09639614143822, −16.24891567812316, −15.98403088927883, −14.89808229340700, −13.83022157944661, −13.31568086666895, −13.04492790263190, −12.11794857015006, −10.88060483167720, −10.51441006473715, −9.386787693249345, −9.078082545589599, −7.220756810595147, −6.372037189139258, −5.946071488121873, −5.263873520096407, −3.783153759641863, −2.779013424568809, −1.387832674516496, 1.387832674516496, 2.779013424568809, 3.783153759641863, 5.263873520096407, 5.946071488121873, 6.372037189139258, 7.220756810595147, 9.078082545589599, 9.386787693249345, 10.51441006473715, 10.88060483167720, 12.11794857015006, 13.04492790263190, 13.31568086666895, 13.83022157944661, 14.89808229340700, 15.98403088927883, 16.24891567812316, 17.09639614143822, 17.85556944710945, 18.48056168104660, 19.25175833806691

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