Properties

Degree 2
Conductor $ 2 \cdot 53 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s − 4·5-s + 6-s − 8-s − 2·9-s + 4·10-s − 4·11-s − 12-s + 13-s + 4·15-s + 16-s + 5·17-s + 2·18-s − 7·19-s − 4·20-s + 4·22-s + 23-s + 24-s + 11·25-s − 26-s + 5·27-s + 5·29-s − 4·30-s − 4·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.78·5-s + 0.408·6-s − 0.353·8-s − 2/3·9-s + 1.26·10-s − 1.20·11-s − 0.288·12-s + 0.277·13-s + 1.03·15-s + 1/4·16-s + 1.21·17-s + 0.471·18-s − 1.60·19-s − 0.894·20-s + 0.852·22-s + 0.208·23-s + 0.204·24-s + 11/5·25-s − 0.196·26-s + 0.962·27-s + 0.928·29-s − 0.730·30-s − 0.718·31-s − 0.176·32-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  =  1
Selberg data  =  $(2,\ 106,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 + 4 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 15 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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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

−19.73492226388343, −18.94313323904141, −18.25906250965574, −16.96977185022352, −16.33479970599235, −15.46267939902897, −14.65173995037189, −12.81150844312398, −11.91352440056693, −11.12054167983950, −10.32538367723606, −8.438990139611851, −8.000989778913675, −6.648774377588663, −5.028034549126719, −3.274779370828331, 0, 3.274779370828331, 5.028034549126719, 6.648774377588663, 8.000989778913675, 8.438990139611851, 10.32538367723606, 11.12054167983950, 11.91352440056693, 12.81150844312398, 14.65173995037189, 15.46267939902897, 16.33479970599235, 16.96977185022352, 18.25906250965574, 18.94313323904141, 19.73492226388343

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