Properties

Degree 2
Conductor $ 13 \cdot 463 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 3·5-s − 3·7-s − 3·9-s − 11-s + 13-s + 4·16-s + 2·17-s − 19-s − 6·20-s + 2·23-s + 4·25-s + 6·28-s + 5·29-s + 4·31-s − 9·35-s + 6·36-s − 9·37-s + 2·41-s − 4·43-s + 2·44-s − 9·45-s + 2·49-s − 2·52-s + 6·53-s − 3·55-s + 10·59-s + ⋯
L(s)  = 1  − 4-s + 1.34·5-s − 1.13·7-s − 9-s − 0.301·11-s + 0.277·13-s + 16-s + 0.485·17-s − 0.229·19-s − 1.34·20-s + 0.417·23-s + 4/5·25-s + 1.13·28-s + 0.928·29-s + 0.718·31-s − 1.52·35-s + 36-s − 1.47·37-s + 0.312·41-s − 0.609·43-s + 0.301·44-s − 1.34·45-s + 2/7·49-s − 0.277·52-s + 0.824·53-s − 0.404·55-s + 1.30·59-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6019 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6019 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6019\)    =    \(13 \cdot 463\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6019} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 6019,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{13,\;463\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{13,\;463\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad13 \( 1 - T \)
463 \( 1 - T \)
good2 \( 1 + p T^{2} \)
3 \( 1 + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 9 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 7 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−17.63822945301851, −17.25610293018367, −16.72619144993820, −16.08006124615298, −15.30140834914835, −14.45459034214936, −14.08198259852669, −13.44338790161493, −13.19079300728513, −12.44807287278476, −11.84590790761291, −10.77553233471109, −10.08485732348009, −9.878123070986103, −9.012210666550603, −8.693950296214009, −7.918665836298322, −6.728668306992591, −6.261637252325831, −5.448842056956229, −5.175725614249083, −3.972069166250087, −3.136109002470584, −2.513262400310362, −1.195447528018953, 0, 1.195447528018953, 2.513262400310362, 3.136109002470584, 3.972069166250087, 5.175725614249083, 5.448842056956229, 6.261637252325831, 6.728668306992591, 7.918665836298322, 8.693950296214009, 9.012210666550603, 9.878123070986103, 10.08485732348009, 10.77553233471109, 11.84590790761291, 12.44807287278476, 13.19079300728513, 13.44338790161493, 14.08198259852669, 14.45459034214936, 15.30140834914835, 16.08006124615298, 16.72619144993820, 17.25610293018367, 17.63822945301851

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