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-s − 2·3-s − 4-s − 5-s − 2·6-s − 2·7-s − 3·8-s + 9-s − 10-s + 2·12-s − 2·13-s − 2·14-s + 2·15-s − 16-s + 2·17-s + 18-s + 2·19-s + 20-s + 4·21-s − 8·23-s + 6·24-s + 25-s − 2·26-s + 4·27-s + 2·28-s + 2·29-s + 2·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s − 1/2·4-s − 0.447·5-s − 0.816·6-s − 0.755·7-s − 1.06·8-s + 1/3·9-s − 0.316·10-s + 0.577·12-s − 0.554·13-s − 0.534·14-s + 0.516·15-s − 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.458·19-s + 0.223·20-s + 0.872·21-s − 1.66·23-s + 1.22·24-s + 1/5·25-s − 0.392·26-s + 0.769·27-s + 0.377·28-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 \( 1 - T + p T^{2} \)
3 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 18 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−19.24955928010571, −18.17407997303286, −17.66970448205745, −16.50374514781415, −16.00499396643651, −14.77812754600729, −13.97959048381603, −12.80970277722265, −12.22791693199163, −11.49667671710561, −10.23148599364935, −9.307883159759705, −7.875138191801485, −6.435706370459347, −5.618198707839457, −4.577869434942991, −3.307155901584770, 0, 3.307155901584770, 4.577869434942991, 5.618198707839457, 6.435706370459347, 7.875138191801485, 9.307883159759705, 10.23148599364935, 11.49667671710561, 12.22791693199163, 12.80970277722265, 13.97959048381603, 14.77812754600729, 16.00499396643651, 16.50374514781415, 17.66970448205745, 18.17407997303286, 19.24955928010571

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