Properties

Degree 2
Conductor $ 5 \cdot 13 $
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 − 2·3-s − 4-s − 5-s + 2·6-s − 4·7-s + 3·8-s + 9-s + 10-s + 2·11-s + 2·12-s − 13-s + 4·14-s + 2·15-s − 16-s + 2·17-s − 18-s − 6·19-s + 20-s + 8·21-s − 2·22-s − 6·23-s − 6·24-s + 25-s + 26-s + 4·27-s + 4·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s − 1/2·4-s − 0.447·5-s + 0.816·6-s − 1.51·7-s + 1.06·8-s + 1/3·9-s + 0.316·10-s + 0.603·11-s + 0.577·12-s − 0.277·13-s + 1.06·14-s + 0.516·15-s − 1/4·16-s + 0.485·17-s − 0.235·18-s − 1.37·19-s + 0.223·20-s + 1.74·21-s − 0.426·22-s − 1.25·23-s − 1.22·24-s + 1/5·25-s + 0.196·26-s + 0.769·27-s + 0.755·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(65\)    =    \(5 \cdot 13\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{65} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 65,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;13\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{5,\;13\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad5 \( 1 + T \)
13 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
3 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−19.97119938138594, −19.29648721649122, −18.45907798787699, −17.34056174403762, −16.68142503985162, −15.97849977668049, −14.36541807055163, −12.91463619461872, −12.13731877933107, −10.75277124581318, −9.844976583062747, −8.679150483937959, −7.042317739968156, −5.806775561758201, −4.035508788587775, 0, 4.035508788587775, 5.806775561758201, 7.042317739968156, 8.679150483937959, 9.844976583062747, 10.75277124581318, 12.13731877933107, 12.91463619461872, 14.36541807055163, 15.97849977668049, 16.68142503985162, 17.34056174403762, 18.45907798787699, 19.29648721649122, 19.97119938138594

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