Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 7 $
Sign $0.657 + 0.753i$
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 + (−2.23 − 0.133i)5-s − 3·6-s + (−0.866 + 2.5i)7-s − 0.999i·8-s + (3 − 5.19i)9-s + (1.86 + 1.23i)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 + (−0.998 − 0.0599i)5-s − 1.22·6-s + (−0.327 + 0.944i)7-s − 0.353i·8-s + (1 − 1.73i)9-s + (0.590 + 0.389i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.657 + 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.657 + 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(70\)    =    \(2 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.657 + 0.753i$
motivic weight  =  \(1\)
character  :  $\chi_{70} (9, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 70,\ (\ :1/2),\ 0.657 + 0.753i)$
$L(1)$  $\approx$  $0.829871 - 0.377410i$
$L(\frac12)$  $\approx$  $0.829871 - 0.377410i$
$L(\frac{3}{2})$   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 \( 1 + (2.23 + 0.133i)T \)
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

−14.71885130992687241874749823264, −13.27839179520853497075201283808, −12.43094751067837540466978761204, −11.47367853440099023350187217876, −9.592427995643476242217597018235, −8.630824227271925815828165466574, −7.939813405285696141072269179497, −6.71260132864708786293101035561, −3.72196179733765636182665178792, −2.27287846813021968208195845828, 3.19474381889672672190380026752, 4.48479960410171582894100632999, 7.12503350874377073437162780492, 8.047881414401046045353644578488, 9.011912729514471905709771583413, 10.13514801880762143898798322353, 11.04772205434518895150237933479, 12.98623277872393056267018527727, 14.19277905824408766246309026195, 14.99126075687247622639054563050

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