Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 7 $
Sign $0.991 - 0.126i$
Motivic weight 1
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 + (1 − 1.73i)3-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s + 1.99·6-s + (−2 + 1.73i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (0.499 − 0.866i)10-s + (−1.5 + 2.59i)11-s + (0.999 + 1.73i)12-s − 13-s + (−2.5 − 0.866i)14-s − 1.99·15-s + (−0.5 − 0.866i)16-s + (3 − 5.19i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.577 − 0.999i)3-s + (−0.249 + 0.433i)4-s + (−0.223 − 0.387i)5-s + 0.816·6-s + (−0.755 + 0.654i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.158 − 0.273i)10-s + (−0.452 + 0.783i)11-s + (0.288 + 0.499i)12-s − 0.277·13-s + (−0.668 − 0.231i)14-s − 0.516·15-s + (−0.125 − 0.216i)16-s + (0.727 − 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(70\)    =    \(2 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.991 - 0.126i$
motivic weight  =  \(1\)
character  :  $\chi_{70} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 70,\ (\ :1/2),\ 0.991 - 0.126i)$
$L(1)$  $\approx$  $1.11281 + 0.0706192i$
$L(\frac12)$  $\approx$  $1.11281 + 0.0706192i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (2 - 1.73i)T \)
good3 \( 1 + (-1 + 1.73i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.5 + 7.79i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.5 - 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + (4.5 + 7.79i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.5 - 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-2 + 3.46i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5 - 8.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−14.57781183609616189848107520147, −13.68579527590435481843316327750, −12.50121732159152632150920036533, −12.24788227275347666382833646586, −9.948938616610033244479183380716, −8.617441261340900804509559804041, −7.60422602125420421213833691831, −6.55221263625625154346273123197, −4.92336720086814895226941027035, −2.73046000169629407206692505395, 3.20395034731392218686573802636, 4.06859797742588904077155461716, 5.95989562751509034618273960297, 7.84833271908281547941958143290, 9.433724438513604545475250100804, 10.19095008837039158422829506066, 11.12291794042038496292391479291, 12.59224917444173072857038973396, 13.71089637602596785375825156220, 14.61777141842794189768454698025

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