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 + 3·5-s − 2·6-s + 2·7-s + 8-s + 9-s + 3·10-s − 3·11-s − 2·12-s − 4·13-s + 2·14-s − 6·15-s + 16-s + 3·17-s + 18-s − 4·19-s + 3·20-s − 4·21-s − 3·22-s − 9·23-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 + 1.34·5-s − 0.816·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.948·10-s − 0.904·11-s − 0.577·12-s − 1.10·13-s + 0.534·14-s − 1.54·15-s + 1/4·16-s + 0.727·17-s + 0.235·18-s − 0.917·19-s + 0.670·20-s − 0.872·21-s − 0.639·22-s − 1.87·23-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  =  $(2,\ 106,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.261927686$
$L(\frac12)$  $\approx$  $1.261927686$
$L(\frac{3}{2})$   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 \( 1 - 3 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + p T^{2} \)
59 \( 1 - 15 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−19.52727953858580, −18.07806930061466, −17.55712864223429, −16.84957304008628, −15.77710377985591, −14.44269412335975, −13.83873739856829, −12.55269108860988, −11.88769378557974, −10.57831280853927, −10.01935142386492, −8.069092544842463, −6.486154299697881, −5.559119618409655, −4.809523451567514, −2.251077140735362, 2.251077140735362, 4.809523451567514, 5.559119618409655, 6.486154299697881, 8.069092544842463, 10.01935142386492, 10.57831280853927, 11.88769378557974, 12.55269108860988, 13.83873739856829, 14.44269412335975, 15.77710377985591, 16.84957304008628, 17.55712864223429, 18.07806930061466, 19.52727953858580

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