Properties

Degree 2
Conductor 61
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 − 3·5-s + 2·6-s + 7-s + 3·8-s + 9-s + 3·10-s − 5·11-s + 2·12-s + 13-s − 14-s + 6·15-s − 16-s + 4·17-s − 18-s − 4·19-s + 3·20-s − 2·21-s + 5·22-s − 9·23-s − 6·24-s + 4·25-s − 26-s + 4·27-s − 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 + 0.377·7-s + 1.06·8-s + 1/3·9-s + 0.948·10-s − 1.50·11-s + 0.577·12-s + 0.277·13-s − 0.267·14-s + 1.54·15-s − 1/4·16-s + 0.970·17-s − 0.235·18-s − 0.917·19-s + 0.670·20-s − 0.436·21-s + 1.06·22-s − 1.87·23-s − 1.22·24-s + 4/5·25-s − 0.196·26-s + 0.769·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(61\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{61} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 61,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 61$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 61$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad61 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−19.34995017826438, −18.41698491619987, −17.86848844698157, −16.61700756854240, −16.06659766840717, −14.73281781004714, −13.13927303138003, −11.99765290224431, −11.01835010861511, −10.14754439690119, −8.287570693733702, −7.660397633337711, −5.601576120880783, −4.256669155871595, 0, 4.256669155871595, 5.601576120880783, 7.660397633337711, 8.287570693733702, 10.14754439690119, 11.01835010861511, 11.99765290224431, 13.13927303138003, 14.73281781004714, 16.06659766840717, 16.61700756854240, 17.86848844698157, 18.41698491619987, 19.34995017826438

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