Properties

Degree 2
Conductor 67
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 2·3-s + 2·4-s + 2·5-s − 4·6-s − 2·7-s + 9-s + 4·10-s − 4·11-s − 4·12-s + 2·13-s − 4·14-s − 4·15-s − 4·16-s + 3·17-s + 2·18-s + 7·19-s + 4·20-s + 4·21-s − 8·22-s + 9·23-s − 25-s + 4·26-s + 4·27-s − 4·28-s − 5·29-s − 8·30-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 4-s + 0.894·5-s − 1.63·6-s − 0.755·7-s + 1/3·9-s + 1.26·10-s − 1.20·11-s − 1.15·12-s + 0.554·13-s − 1.06·14-s − 1.03·15-s − 16-s + 0.727·17-s + 0.471·18-s + 1.60·19-s + 0.894·20-s + 0.872·21-s − 1.70·22-s + 1.87·23-s − 1/5·25-s + 0.784·26-s + 0.769·27-s − 0.755·28-s − 0.928·29-s − 1.46·30-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(67\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{67} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 67,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.273770036$
$L(\frac12)$  $\approx$  $1.273770036$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 67$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 67$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad67 \( 1 - T \)
good2 \( 1 - p T + p T^{2} \)
3 \( 1 + 2 T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 - 9 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 7 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−18.59770497184902, −17.88281308707341, −16.62153978016763, −15.89170441818060, −14.62645342322259, −13.35065815414322, −12.96065758485635, −11.74279646718555, −10.71433657084450, −9.394927873325611, −7.006765778294681, −5.556806967144734, −5.422087600288199, −3.189874361727484, 3.189874361727484, 5.422087600288199, 5.556806967144734, 7.006765778294681, 9.394927873325611, 10.71433657084450, 11.74279646718555, 12.96065758485635, 13.35065815414322, 14.62645342322259, 15.89170441818060, 16.62153978016763, 17.88281308707341, 18.59770497184902

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