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 − 3·7-s − 8-s + 9-s + 10-s − 11-s − 12-s + 3·14-s + 15-s + 16-s − 17-s − 18-s − 4·19-s − 20-s + 3·21-s + 22-s − 23-s + 24-s + 25-s − 27-s − 3·28-s − 6·29-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 − 1.13·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s − 0.288·12-s + 0.801·14-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 + 0.654·21-s + 0.213·22-s − 0.208·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s − 0.566·28-s − 1.11·29-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 \( 1 + 3 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 - 11 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−14.37700702660694, −13.36032072419543, −13.04449640027538, −12.69384747484839, −12.16258868981871, −11.59738327364488, −10.99481074875348, −10.84325457751582, −10.08563147990582, −9.749963924047997, −9.168563593983003, −8.773703976361651, −7.979023079397595, −7.658717614675143, −7.024655288384513, −6.385984920737771, −6.273875860880180, −5.458499500277860, −4.893169686157925, −4.109039054132729, −3.587569595883237, −2.996984750792540, −2.207437992673308, −1.585490617118284, −0.5256757219549161, 0, 0.5256757219549161, 1.585490617118284, 2.207437992673308, 2.996984750792540, 3.587569595883237, 4.109039054132729, 4.893169686157925, 5.458499500277860, 6.273875860880180, 6.385984920737771, 7.024655288384513, 7.658717614675143, 7.979023079397595, 8.773703976361651, 9.168563593983003, 9.749963924047997, 10.08563147990582, 10.84325457751582, 10.99481074875348, 11.59738327364488, 12.16258868981871, 12.69384747484839, 13.04449640027538, 13.36032072419543, 14.37700702660694

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