Properties

Degree 2
Conductor 37
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·4-s − 7-s − 2·9-s + 3·11-s − 2·12-s − 4·13-s + 4·16-s + 6·17-s + 2·19-s − 21-s + 6·23-s − 5·25-s − 5·27-s + 2·28-s − 6·29-s − 4·31-s + 3·33-s + 4·36-s + 37-s − 4·39-s − 9·41-s + 8·43-s − 6·44-s + 3·47-s + 4·48-s − 6·49-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s − 0.377·7-s − 2/3·9-s + 0.904·11-s − 0.577·12-s − 1.10·13-s + 16-s + 1.45·17-s + 0.458·19-s − 0.218·21-s + 1.25·23-s − 25-s − 0.962·27-s + 0.377·28-s − 1.11·29-s − 0.718·31-s + 0.522·33-s + 2/3·36-s + 0.164·37-s − 0.640·39-s − 1.40·41-s + 1.21·43-s − 0.904·44-s + 0.437·47-s + 0.577·48-s − 6/7·49-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(37\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{37} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 37,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.7256810619$
$L(\frac12)$  $\approx$  $0.7256810619$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 37$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 37$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad37 \( 1 - T \)
good2 \( 1 + p T^{2} \)
3 \( 1 - T + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−19.49147182698229, −18.94371679359437, −17.43722746569670, −16.72216150673137, −14.80595468440373, −14.26876170526783, −13.04504337075354, −11.79612239530835, −9.819966817497619, −9.032345851694468, −7.599111770673714, −5.449734162154715, −3.509102943404796, 3.509102943404796, 5.449734162154715, 7.599111770673714, 9.032345851694468, 9.819966817497619, 11.79612239530835, 13.04504337075354, 14.26876170526783, 14.80595468440373, 16.72216150673137, 17.43722746569670, 18.94371679359437, 19.49147182698229

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