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 + 3·7-s − 8-s + 9-s + 10-s + 5·11-s − 12-s − 3·14-s + 15-s + 16-s + 17-s − 18-s + 2·19-s − 20-s − 3·21-s − 5·22-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 + 1.50·11-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 + 0.458·19-s − 0.223·20-s − 0.654·21-s − 1.06·22-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  =  0
Selberg data  =  $(2,\ 86190,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.868937825$
$L(\frac12)$  $\approx$  $1.868937825$
$L(\frac{3}{2})$   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 \( 1 - 5 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 9 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 13 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 5 T + p T^{2} \)
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

−14.12245916430373, −13.40813829161192, −12.72965474882581, −12.18942884177472, −11.79246976491478, −11.35802643604995, −11.07888096902903, −10.61205582915494, −9.692896841548358, −9.539187768558390, −8.949102085130037, −8.247330081268834, −7.894934712577854, −7.422197392544798, −6.808123438138111, −6.282809387717435, −5.814287044392020, −4.962049494541777, −4.687632139097167, −3.774086212201301, −3.546095465514780, −2.417473294190650, −1.745973151155495, −1.176673892040080, −0.5840500021236037, 0.5840500021236037, 1.176673892040080, 1.745973151155495, 2.417473294190650, 3.546095465514780, 3.774086212201301, 4.687632139097167, 4.962049494541777, 5.814287044392020, 6.282809387717435, 6.808123438138111, 7.422197392544798, 7.894934712577854, 8.247330081268834, 8.949102085130037, 9.539187768558390, 9.692896841548358, 10.61205582915494, 11.07888096902903, 11.35802643604995, 11.79246976491478, 12.18942884177472, 12.72965474882581, 13.40813829161192, 14.12245916430373

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