Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 19 $
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 − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s − 2·9-s + 10-s − 12-s − 3·13-s + 14-s + 15-s + 16-s − 7·17-s + 2·18-s − 19-s − 20-s + 21-s − 5·23-s + 24-s + 25-s + 3·26-s + 5·27-s − 28-s − 5·29-s − 30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s − 2/3·9-s + 0.316·10-s − 0.288·12-s − 0.832·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 1.69·17-s + 0.471·18-s − 0.229·19-s − 0.223·20-s + 0.218·21-s − 1.04·23-s + 0.204·24-s + 1/5·25-s + 0.588·26-s + 0.962·27-s − 0.188·28-s − 0.928·29-s − 0.182·30-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(190\)    =    \(2 \cdot 5 \cdot 19\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{190} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 190,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;19\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;19\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
5 \( 1 + T \)
19 \( 1 + T \)
good3 \( 1 + T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 7 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−19.78233233757624, −19.35555674871269, −18.17589639595404, −17.47618140398713, −16.80188353609426, −15.89035089622342, −15.15684094695180, −14.00483179906586, −12.75562982672695, −11.80119378341821, −11.14134483144775, −10.13250645401623, −9.052009488114564, −8.079561289589277, −6.885518213856033, −5.936215018091487, −4.420871974151108, −2.580321940532615, 0, 2.580321940532615, 4.420871974151108, 5.936215018091487, 6.885518213856033, 8.079561289589277, 9.052009488114564, 10.13250645401623, 11.14134483144775, 11.80119378341821, 12.75562982672695, 14.00483179906586, 15.15684094695180, 15.89035089622342, 16.80188353609426, 17.47618140398713, 18.17589639595404, 19.35555674871269, 19.78233233757624

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