Properties

Degree 2
Conductor $ 19 \cdot 1279 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 3

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s − 4-s − 2·5-s + 2·6-s − 4·7-s + 3·8-s + 9-s + 2·10-s − 5·11-s + 2·12-s − 13-s + 4·14-s + 4·15-s − 16-s − 7·17-s − 18-s − 19-s + 2·20-s + 8·21-s + 5·22-s − 6·23-s − 6·24-s − 25-s + 26-s + 4·27-s + 4·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s − 1/2·4-s − 0.894·5-s + 0.816·6-s − 1.51·7-s + 1.06·8-s + 1/3·9-s + 0.632·10-s − 1.50·11-s + 0.577·12-s − 0.277·13-s + 1.06·14-s + 1.03·15-s − 1/4·16-s − 1.69·17-s − 0.235·18-s − 0.229·19-s + 0.447·20-s + 1.74·21-s + 1.06·22-s − 1.25·23-s − 1.22·24-s − 1/5·25-s + 0.196·26-s + 0.769·27-s + 0.755·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 24301 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 24301 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(24301\)    =    \(19 \cdot 1279\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{24301} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  3
Selberg data  =  $(2,\ 24301,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{19,\;1279\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{19,\;1279\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad19 \( 1 + T \)
1279 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 5 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−16.22061354055168, −15.95683664675589, −15.40571477013031, −14.86983841573863, −13.73105337931760, −13.46464575534315, −12.77944315215153, −12.58898104343362, −11.76072054445982, −11.23977646891696, −10.72188616645509, −10.14711236144525, −9.875914551092611, −8.994827342421628, −8.590543604108184, −7.877646893311826, −7.273742488299939, −6.834289713977547, −5.933837986552611, −5.623109961150586, −4.694094312033944, −4.295227591662289, −3.503627521261095, −2.695409970021237, −1.710188776452033, 0, 0, 0, 1.710188776452033, 2.695409970021237, 3.503627521261095, 4.295227591662289, 4.694094312033944, 5.623109961150586, 5.933837986552611, 6.834289713977547, 7.273742488299939, 7.877646893311826, 8.590543604108184, 8.994827342421628, 9.875914551092611, 10.14711236144525, 10.72188616645509, 11.23977646891696, 11.76072054445982, 12.58898104343362, 12.77944315215153, 13.46464575534315, 13.73105337931760, 14.86983841573863, 15.40571477013031, 15.95683664675589, 16.22061354055168

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