Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 7 $
Sign $0.208 - 0.978i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (−2.59 + 1.5i)3-s + (0.499 + 0.866i)4-s + (1.23 + 1.86i)5-s − 3·6-s + (0.866 − 2.5i)7-s + 0.999i·8-s + (3 − 5.19i)9-s + (0.133 + 2.23i)10-s + (−2.59 − 1.50i)12-s − 2i·13-s + (2 − 1.73i)14-s + (−6 − 3i)15-s + (−0.5 + 0.866i)16-s + (1.73 − i)17-s + (5.19 − 3i)18-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−1.49 + 0.866i)3-s + (0.249 + 0.433i)4-s + (0.550 + 0.834i)5-s − 1.22·6-s + (0.327 − 0.944i)7-s + 0.353i·8-s + (1 − 1.73i)9-s + (0.0423 + 0.705i)10-s + (−0.749 − 0.433i)12-s − 0.554i·13-s + (0.534 − 0.462i)14-s + (−1.54 − 0.774i)15-s + (−0.125 + 0.216i)16-s + (0.420 − 0.242i)17-s + (1.22 − 0.707i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.208 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.208 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(70\)    =    \(2 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.208 - 0.978i$
motivic weight  =  \(1\)
character  :  $\chi_{70} (9, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 70,\ (\ :1/2),\ 0.208 - 0.978i)$
$L(1)$  $\approx$  $0.708696 + 0.573475i$
$L(\frac12)$  $\approx$  $0.708696 + 0.573475i$
$L(\frac{3}{2})$   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 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 + (-1.23 - 1.86i)T \)
7 \( 1 + (-0.866 + 2.5i)T \)
good3 \( 1 + (2.59 - 1.5i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + (-1.73 + i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 + (5 + 8.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.92 + 4i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 - 5iT - 43T^{2} \)
47 \( 1 + (-6.92 - 4i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.19 - 3i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1 - 1.73i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.5 + 7.79i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.06 + 3.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (8.66 - 5i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5 - 8.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9iT - 83T^{2} \)
89 \( 1 + (3.5 - 6.06i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−15.01378994561969518513406450315, −14.04688400236946830188324989970, −12.74016830980429047250725311396, −11.39444897057865450990211041701, −10.70567023931131578997706477842, −9.806660943830376443911478166377, −7.42171420125352958769405301091, −6.21924130930249366732776460977, −5.22625661068966867787152024155, −3.81770243724999369473839296429, 1.72010951584794250908056878961, 4.94425525517896381904069506670, 5.67341944065894258133949738608, 6.84056006356308735358610205757, 8.745418404231952740762255374242, 10.39247182324690179314206702678, 11.61538378737480570837057754073, 12.28117800541804475435058764838, 12.97293312071078694527113425833, 14.12965696730763055122141290614

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