Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3·3-s + 2·4-s − 2·5-s + 6·6-s − 7-s + 6·9-s + 4·10-s − 5·11-s − 6·12-s − 2·13-s + 2·14-s + 6·15-s − 4·16-s − 12·18-s − 4·20-s + 3·21-s + 10·22-s + 2·23-s − 25-s + 4·26-s − 9·27-s − 2·28-s + 6·29-s − 12·30-s − 4·31-s + 8·32-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.73·3-s + 4-s − 0.894·5-s + 2.44·6-s − 0.377·7-s + 2·9-s + 1.26·10-s − 1.50·11-s − 1.73·12-s − 0.554·13-s + 0.534·14-s + 1.54·15-s − 16-s − 2.82·18-s − 0.894·20-s + 0.654·21-s + 2.13·22-s + 0.417·23-s − 1/5·25-s + 0.784·26-s − 1.73·27-s − 0.377·28-s + 1.11·29-s − 2.19·30-s − 0.718·31-s + 1.41·32-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  =  1
Selberg data  =  $(2,\ 37,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−19.81482224536338, −18.78719562466392, −18.06365420291071, −17.14169364801487, −16.19201741687448, −15.60385787320432, −12.95838641388285, −11.75732472284978, −10.77513816254080, −9.933098353605352, −8.014330807872879, −6.870391216954432, −5.003170014006659, 0, 5.003170014006659, 6.870391216954432, 8.014330807872879, 9.933098353605352, 10.77513816254080, 11.75732472284978, 12.95838641388285, 15.60385787320432, 16.19201741687448, 17.14169364801487, 18.06365420291071, 18.78719562466392, 19.81482224536338

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