Properties

Degree 2
Conductor $ 2^{2} \cdot 5 \cdot 7 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 5-s − 7-s + 6·9-s − 5·11-s − 3·13-s − 3·15-s − 17-s + 6·19-s − 3·21-s + 6·23-s + 25-s + 9·27-s − 9·29-s − 4·31-s − 15·33-s + 35-s + 2·37-s − 9·39-s − 4·41-s + 10·43-s − 6·45-s − 47-s + 49-s − 3·51-s + 4·53-s + 5·55-s + ⋯
L(s)  = 1  + 1.73·3-s − 0.447·5-s − 0.377·7-s + 2·9-s − 1.50·11-s − 0.832·13-s − 0.774·15-s − 0.242·17-s + 1.37·19-s − 0.654·21-s + 1.25·23-s + 1/5·25-s + 1.73·27-s − 1.67·29-s − 0.718·31-s − 2.61·33-s + 0.169·35-s + 0.328·37-s − 1.44·39-s − 0.624·41-s + 1.52·43-s − 0.894·45-s − 0.145·47-s + 1/7·49-s − 0.420·51-s + 0.549·53-s + 0.674·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{140} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 140,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.555511632$
$L(\frac12)$  $\approx$  $1.555511632$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;7\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
5 \( 1 + T \)
7 \( 1 + T \)
good3 \( 1 - p T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−19.69522344795471, −18.82925267581958, −18.26201889638481, −16.67514292178483, −15.58696439469901, −15.15455027541741, −14.13165716056992, −13.22652980823547, −12.58088516701521, −10.97393679911059, −9.743659591392654, −9.025895505037625, −7.737317764551611, −7.339864661387265, −5.112482776824541, −3.514750050353024, −2.511632173168903, 2.511632173168903, 3.514750050353024, 5.112482776824541, 7.339864661387265, 7.737317764551611, 9.025895505037625, 9.743659591392654, 10.97393679911059, 12.58088516701521, 13.22652980823547, 14.13165716056992, 15.15455027541741, 15.58696439469901, 16.67514292178483, 18.26201889638481, 18.82925267581958, 19.69522344795471

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