Properties

Degree 2
Conductor $ 5 \cdot 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-s + 2·4-s − 5-s − 2·6-s − 5·7-s − 2·9-s + 2·10-s + 3·11-s + 2·12-s − 2·13-s + 10·14-s − 15-s − 4·16-s − 4·17-s + 4·18-s − 4·19-s − 2·20-s − 5·21-s − 6·22-s − 2·23-s + 25-s + 4·26-s − 5·27-s − 10·28-s + 2·29-s + 2·30-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s + 4-s − 0.447·5-s − 0.816·6-s − 1.88·7-s − 2/3·9-s + 0.632·10-s + 0.904·11-s + 0.577·12-s − 0.554·13-s + 2.67·14-s − 0.258·15-s − 16-s − 0.970·17-s + 0.942·18-s − 0.917·19-s − 0.447·20-s − 1.09·21-s − 1.27·22-s − 0.417·23-s + 1/5·25-s + 0.784·26-s − 0.962·27-s − 1.88·28-s + 0.371·29-s + 0.365·30-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(185\)    =    \(5 \cdot 37\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{185} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 185,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;37\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{5,\;37\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad5 \( 1 + T \)
37 \( 1 + T \)
good2 \( 1 + p T + p T^{2} \)
3 \( 1 - T + p T^{2} \)
7 \( 1 + 5 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 11 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
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

−19.93162813323368, −19.28815215716707, −18.74802666767931, −17.42156218848558, −16.86372628061703, −16.03565974922532, −15.23041458522032, −14.02701995405544, −13.03246348777728, −11.96118951087324, −10.80170433061065, −9.758331603941689, −9.108040044433494, −8.403685986159717, −7.116351026476220, −6.310904606715823, −3.994595524652362, −2.568213157743029, 0, 2.568213157743029, 3.994595524652362, 6.310904606715823, 7.116351026476220, 8.403685986159717, 9.108040044433494, 9.758331603941689, 10.80170433061065, 11.96118951087324, 13.03246348777728, 14.02701995405544, 15.23041458522032, 16.03565974922532, 16.86372628061703, 17.42156218848558, 18.74802666767931, 19.28815215716707, 19.93162813323368

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