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 0

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 − 4·19-s − 20-s − 2·21-s + 22-s − 9·23-s + 24-s + 25-s − 27-s + 2·28-s − 7·29-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 − 0.917·19-s − 0.223·20-s − 0.436·21-s + 0.213·22-s − 1.87·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s + 0.377·28-s − 1.29·29-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  =  0
Selberg data  =  $(2,\ 86190,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.2395745092$
$L(\frac12)$  $\approx$  $0.2395745092$
$L(\frac{3}{2})$   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 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + 9 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−14.01032260583921, −13.39229822249297, −12.69942470605036, −12.34335041378798, −11.71674506353158, −11.49658872047072, −10.95723763488501, −10.30539887345573, −10.14303834318650, −9.512862953965607, −8.609499932387917, −8.491723194671158, −7.823882747723542, −7.458677420186378, −6.862196270916642, −6.161208181419973, −5.846192059546688, −5.065076173142904, −4.592412672840850, −3.924681575997525, −3.376599158876892, −2.398350571092888, −1.861541133940319, −1.258190268833380, −0.1877001892775763, 0.1877001892775763, 1.258190268833380, 1.861541133940319, 2.398350571092888, 3.376599158876892, 3.924681575997525, 4.592412672840850, 5.065076173142904, 5.846192059546688, 6.161208181419973, 6.862196270916642, 7.458677420186378, 7.823882747723542, 8.491723194671158, 8.609499932387917, 9.512862953965607, 10.14303834318650, 10.30539887345573, 10.95723763488501, 11.49658872047072, 11.71674506353158, 12.34335041378798, 12.69942470605036, 13.39229822249297, 14.01032260583921

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