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 + 2·7-s − 8-s + 9-s − 10-s − 11-s − 12-s − 2·14-s − 15-s + 16-s + 17-s − 18-s + 2·19-s + 20-s − 2·21-s + 22-s − 3·23-s + 24-s + 25-s − 27-s + 2·28-s + 3·29-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 + 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.301·11-s − 0.288·12-s − 0.534·14-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 − 0.436·21-s + 0.213·22-s − 0.625·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s + 0.377·28-s + 0.557·29-s + 0.182·30-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 \( 1 - 2 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 7 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
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

−14.13106489594005, −13.70183560894537, −13.19578476561548, −12.45076416468264, −12.13187481818464, −11.56944954326914, −11.20403726683414, −10.57876225711377, −10.21345074640515, −9.754788913755682, −9.260391340633209, −8.514994965818668, −8.136541142786941, −7.728483650308826, −6.886822737248374, −6.704060531564630, −5.881201614913843, −5.485236271627970, −4.896642942521034, −4.403764604712476, −3.491567628772660, −2.906418775568593, −2.046669331919912, −1.592029990771539, −0.8911367067023357, 0, 0.8911367067023357, 1.592029990771539, 2.046669331919912, 2.906418775568593, 3.491567628772660, 4.403764604712476, 4.896642942521034, 5.485236271627970, 5.881201614913843, 6.704060531564630, 6.886822737248374, 7.728483650308826, 8.136541142786941, 8.514994965818668, 9.260391340633209, 9.754788913755682, 10.21345074640515, 10.57876225711377, 11.20403726683414, 11.56944954326914, 12.13187481818464, 12.45076416468264, 13.19578476561548, 13.70183560894537, 14.13106489594005

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