Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 11 \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 + 4-s − 5-s − 4·7-s + 8-s − 10-s + 11-s + 2·13-s − 4·14-s + 16-s − 6·17-s + 19-s − 20-s + 22-s + 25-s + 2·26-s − 4·28-s + 6·29-s − 4·31-s + 32-s − 6·34-s + 4·35-s + 2·37-s + 38-s − 40-s − 6·41-s + 8·43-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.51·7-s + 0.353·8-s − 0.316·10-s + 0.301·11-s + 0.554·13-s − 1.06·14-s + 1/4·16-s − 1.45·17-s + 0.229·19-s − 0.223·20-s + 0.213·22-s + 1/5·25-s + 0.392·26-s − 0.755·28-s + 1.11·29-s − 0.718·31-s + 0.176·32-s − 1.02·34-s + 0.676·35-s + 0.328·37-s + 0.162·38-s − 0.158·40-s − 0.937·41-s + 1.21·43-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 18810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 18810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(18810\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11 \cdot 19\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{18810} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 18810,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;11,\;19\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;11,\;19\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 + T \)
11 \( 1 - T \)
19 \( 1 - T \)
good7 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−15.85169208656858, −15.62565274721755, −14.97561350530465, −14.36719175976381, −13.59649172891690, −13.35067723275467, −12.79573845100639, −12.24200514583717, −11.75276047871716, −11.03950082738435, −10.62981663378457, −9.887760224835029, −9.293908299170770, −8.724119077096563, −8.094818515806428, −7.136381944784705, −6.761350367837597, −6.308324859355992, −5.629334605140584, −4.848588113048141, −3.983635098788444, −3.744998047381393, −2.867684304597296, −2.309624841419279, −1.069531560241763, 0, 1.069531560241763, 2.309624841419279, 2.867684304597296, 3.744998047381393, 3.983635098788444, 4.848588113048141, 5.629334605140584, 6.308324859355992, 6.761350367837597, 7.136381944784705, 8.094818515806428, 8.724119077096563, 9.293908299170770, 9.887760224835029, 10.62981663378457, 11.03950082738435, 11.75276047871716, 12.24200514583717, 12.79573845100639, 13.35067723275467, 13.59649172891690, 14.36719175976381, 14.97561350530465, 15.62565274721755, 15.85169208656858

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