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 + 12-s + 2·14-s − 15-s + 16-s − 17-s − 18-s − 6·19-s − 20-s − 2·21-s − 24-s + 25-s + 27-s − 2·28-s + 6·29-s + 30-s + 4·31-s − 32-s + 34-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 + 0.288·12-s + 0.534·14-s − 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 1.37·19-s − 0.223·20-s − 0.436·21-s − 0.204·24-s + 1/5·25-s + 0.192·27-s − 0.377·28-s + 1.11·29-s + 0.182·30-s + 0.718·31-s − 0.176·32-s + 0.171·34-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  =  $(2,\ 86190,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.456190736$
$L(\frac12)$  $\approx$  $1.456190736$
$L(\frac{3}{2})$   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 \( 1 + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + p T^{2} \)
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

−14.03657211877151, −13.32087509654560, −12.82659929094766, −12.55429632723809, −11.87680083099684, −11.43916330354576, −10.74887444169175, −10.34563888955421, −9.964225072191776, −9.280132793782286, −8.893880067630880, −8.406204036723643, −7.985807568765031, −7.401921413961360, −6.755095233593087, −6.483408862237044, −5.870771235860570, −5.042674195098004, −4.291215484996871, −3.912817701730238, −3.174836313535821, −2.542329205001904, −2.168803635912626, −1.123669037683048, −0.4624589667672753, 0.4624589667672753, 1.123669037683048, 2.168803635912626, 2.542329205001904, 3.174836313535821, 3.912817701730238, 4.291215484996871, 5.042674195098004, 5.870771235860570, 6.483408862237044, 6.755095233593087, 7.401921413961360, 7.985807568765031, 8.406204036723643, 8.893880067630880, 9.280132793782286, 9.964225072191776, 10.34563888955421, 10.74887444169175, 11.43916330354576, 11.87680083099684, 12.55429632723809, 12.82659929094766, 13.32087509654560, 14.03657211877151

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