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 + 2·3-s + 4-s + 5-s − 2·6-s − 2·7-s − 8-s + 9-s − 10-s + 5·11-s + 2·12-s − 4·13-s + 2·14-s + 2·15-s + 16-s + 3·17-s − 18-s − 4·19-s + 20-s − 4·21-s − 5·22-s − 3·23-s − 2·24-s − 4·25-s + 4·26-s − 4·27-s − 2·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.447·5-s − 0.816·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.50·11-s + 0.577·12-s − 1.10·13-s + 0.534·14-s + 0.516·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 0.917·19-s + 0.223·20-s − 0.872·21-s − 1.06·22-s − 0.625·23-s − 0.408·24-s − 4/5·25-s + 0.784·26-s − 0.769·27-s − 0.377·28-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  =  $(2,\ 106,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.042161443$
$L(\frac12)$  $\approx$  $1.042161443$
$L(\frac{3}{2})$   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 \( 1 - 2 T + p T^{2} \)
5 \( 1 - T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 7 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 + 7 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 - 17 T + p T^{2} \)
97 \( 1 - 3 T + p T^{2} \)
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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

−19.34399374311650, −19.02735110042505, −17.37105079739618, −16.97867625905007, −15.67747400022163, −14.60280487678243, −14.05035500006717, −12.72262421747467, −11.67583324398273, −9.988687509954687, −9.472281232368603, −8.516972320011435, −7.296233943244949, −6.076335421236883, −3.739950069983357, −2.227718685672233, 2.227718685672233, 3.739950069983357, 6.076335421236883, 7.296233943244949, 8.516972320011435, 9.472281232368603, 9.988687509954687, 11.67583324398273, 12.72262421747467, 14.05035500006717, 14.60280487678243, 15.67747400022163, 16.97867625905007, 17.37105079739618, 19.02735110042505, 19.34399374311650

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