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 2

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 − 4·11-s − 12-s + 15-s + 16-s + 17-s − 18-s − 4·19-s − 20-s + 4·22-s + 24-s + 25-s − 27-s − 2·29-s − 30-s − 8·31-s − 32-s + 4·33-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 − 1.20·11-s − 0.288·12-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 + 0.852·22-s + 0.204·24-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.182·30-s − 1.43·31-s − 0.176·32-s + 0.696·33-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  =  2
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 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−14.63003325815575, −13.91216311564751, −13.16366476041470, −12.88305415131049, −12.36843942683794, −11.91925836594107, −11.23994286997003, −10.85004621407969, −10.58370092735669, −10.02591652744466, −9.355176608291319, −8.992426961941884, −8.223993794154117, −7.844789015265567, −7.483108568980980, −6.753269170524058, −6.374014302468732, −5.645724250092362, −5.156588983087483, −4.671445334348317, −3.795673862483803, −3.349798886023550, −2.488402356100131, −1.930277000935988, −1.155231404685803, 0, 0, 1.155231404685803, 1.930277000935988, 2.488402356100131, 3.349798886023550, 3.795673862483803, 4.671445334348317, 5.156588983087483, 5.645724250092362, 6.374014302468732, 6.753269170524058, 7.483108568980980, 7.844789015265567, 8.223993794154117, 8.992426961941884, 9.355176608291319, 10.02591652744466, 10.58370092735669, 10.85004621407969, 11.23994286997003, 11.91925836594107, 12.36843942683794, 12.88305415131049, 13.16366476041470, 13.91216311564751, 14.63003325815575

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