Properties

Degree 2
Conductor $ 2^{2} \cdot 5 \cdot 7 $
Sign $0.0633 + 0.997i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)5-s − 0.999·6-s + (0.5 + 0.866i)7-s − 0.999·8-s + (0.499 + 0.866i)10-s + (−0.499 + 0.866i)12-s + 0.999·14-s + 0.999·15-s + (−0.5 + 0.866i)16-s + 0.999·20-s + (0.499 − 0.866i)21-s + (−0.5 + 0.866i)23-s + (0.499 + 0.866i)24-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)5-s − 0.999·6-s + (0.5 + 0.866i)7-s − 0.999·8-s + (0.499 + 0.866i)10-s + (−0.499 + 0.866i)12-s + 0.999·14-s + 0.999·15-s + (−0.5 + 0.866i)16-s + 0.999·20-s + (0.499 − 0.866i)21-s + (−0.5 + 0.866i)23-s + (0.499 + 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.0633 + 0.997i$
motivic weight  =  \(0\)
character  :  $\chi_{140} (79, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 140,\ (\ :0),\ 0.0633 + 0.997i)$
$L(\frac{1}{2})$  $\approx$  $0.6380148278$
$L(\frac12)$  $\approx$  $0.6380148278$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;7\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;5,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good3 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T + T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−12.91953457458712685832370021318, −11.89730415182283369245167736241, −11.59390338033710325339197060635, −10.49131899340443749749830472998, −9.186924837552491493319425722300, −7.71848071459985778649623621210, −6.44825492248817334948738652479, −5.42165937945702167287681403977, −3.69435910234721680945246195208, −2.07304643702320871970159871316, 3.98412937626303021390155283110, 4.62317910412855121571655553895, 5.67425283931826133655283997430, 7.29984296400960119785591340565, 8.208014038692492856373607073400, 9.392275456734232582728614569972, 10.67564303383999914924915166349, 11.71208045192582949871815458486, 12.78482575618033582120647928994, 13.72109285787297334621672318409

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