Properties

Degree 2
Conductor $ 2 \cdot 53 $
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 + 6-s − 4·7-s + 8-s − 2·9-s + 12-s + 5·13-s − 4·14-s + 16-s − 3·17-s − 2·18-s − 19-s − 4·21-s + 3·23-s + 24-s − 5·25-s + 5·26-s − 5·27-s − 4·28-s + 9·29-s − 4·31-s + 32-s − 3·34-s − 2·36-s + 5·37-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.51·7-s + 0.353·8-s − 2/3·9-s + 0.288·12-s + 1.38·13-s − 1.06·14-s + 1/4·16-s − 0.727·17-s − 0.471·18-s − 0.229·19-s − 0.872·21-s + 0.625·23-s + 0.204·24-s − 25-s + 0.980·26-s − 0.962·27-s − 0.755·28-s + 1.67·29-s − 0.718·31-s + 0.176·32-s − 0.514·34-s − 1/3·36-s + 0.821·37-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 106 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 106 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(106\)    =    \(2 \cdot 53\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{106} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 106,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.524798432$
$L(\frac12)$  $\approx$  $1.524798432$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;53\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;53\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
53 \( 1 + T \)
good3 \( 1 - T + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−19.67968182528900, −18.86185639935096, −17.53570398478070, −16.26854225612658, −15.71955085565582, −14.64804899760430, −13.48097305287935, −13.15344609438076, −11.81761494303417, −10.69149667428674, −9.381217023375045, −8.357926167105308, −6.726408944918774, −5.851303314971287, −3.900152090179058, −2.810730958874394, 2.810730958874394, 3.900152090179058, 5.851303314971287, 6.726408944918774, 8.357926167105308, 9.381217023375045, 10.69149667428674, 11.81761494303417, 13.15344609438076, 13.48097305287935, 14.64804899760430, 15.71955085565582, 16.26854225612658, 17.53570398478070, 18.86185639935096, 19.67968182528900

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