Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 13^{2} \cdot 17 $
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 − 8-s + 9-s − 10-s + 2·11-s − 12-s − 15-s + 16-s − 17-s − 18-s + 2·19-s + 20-s − 2·22-s − 2·23-s + 24-s + 25-s − 27-s − 2·29-s + 30-s − 32-s − 2·33-s + 34-s + 36-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.603·11-s − 0.288·12-s − 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.458·19-s + 0.223·20-s − 0.426·22-s − 0.417·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.182·30-s − 0.176·32-s − 0.348·33-s + 0.171·34-s + 1/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 86190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(86190\)    =    \(2 \cdot 3 \cdot 5 \cdot 13^{2} \cdot 17\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{86190} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 86190,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−14.20649803977565, −13.65585757297579, −13.04869908543646, −12.68315704597392, −11.95612095989551, −11.70392131415163, −11.09933680852976, −10.72772130154916, −10.11138054696800, −9.625679360690653, −9.321427788983558, −8.687015265517496, −8.165292022213955, −7.464202749629514, −7.150441545332719, −6.353388639248884, −6.137039778100385, −5.541069278736797, −4.837085728895615, −4.313108438885710, −3.532582592829836, −2.926042608814919, −2.086628548341341, −1.554095020708446, −0.8666273163049122, 0, 0.8666273163049122, 1.554095020708446, 2.086628548341341, 2.926042608814919, 3.532582592829836, 4.313108438885710, 4.837085728895615, 5.541069278736797, 6.137039778100385, 6.353388639248884, 7.150441545332719, 7.464202749629514, 8.165292022213955, 8.687015265517496, 9.321427788983558, 9.625679360690653, 10.11138054696800, 10.72772130154916, 11.09933680852976, 11.70392131415163, 11.95612095989551, 12.68315704597392, 13.04869908543646, 13.65585757297579, 14.20649803977565

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