Properties

Degree 2
Conductor 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 − 3·3-s − 4-s + 3·6-s − 4·7-s + 3·8-s + 6·9-s + 3·12-s − 3·13-s + 4·14-s − 16-s − 3·17-s − 6·18-s − 5·19-s + 12·21-s + 7·23-s − 9·24-s − 5·25-s + 3·26-s − 9·27-s + 4·28-s − 7·29-s + 4·31-s − 5·32-s + 3·34-s − 6·36-s + 5·37-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.73·3-s − 1/2·4-s + 1.22·6-s − 1.51·7-s + 1.06·8-s + 2·9-s + 0.866·12-s − 0.832·13-s + 1.06·14-s − 1/4·16-s − 0.727·17-s − 1.41·18-s − 1.14·19-s + 2.61·21-s + 1.45·23-s − 1.83·24-s − 25-s + 0.588·26-s − 1.73·27-s + 0.755·28-s − 1.29·29-s + 0.718·31-s − 0.883·32-s + 0.514·34-s − 36-s + 0.821·37-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

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

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 53$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 53$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad53 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
3 \( 1 + p T + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−19.55130022540896, −18.96307855314917, −17.87397635420920, −16.99882325261332, −16.56634537954273, −15.32142426611891, −13.21929603222722, −12.62142615234457, −11.17082899287976, −10.14893340042306, −9.264238826193943, −7.207369479992710, −6.043478899410247, −4.508628350682962, 0, 4.508628350682962, 6.043478899410247, 7.207369479992710, 9.264238826193943, 10.14893340042306, 11.17082899287976, 12.62142615234457, 13.21929603222722, 15.32142426611891, 16.56634537954273, 16.99882325261332, 17.87397635420920, 18.96307855314917, 19.55130022540896

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