Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.999 + 0.00318i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − i)2-s + (−2.84 − 0.947i)3-s + 2i·4-s + (−4.64 − 1.84i)5-s + (1.89 + 3.79i)6-s + (−6.25 + 3.13i)7-s + (2 − 2i)8-s + (7.20 + 5.39i)9-s + (2.79 + 6.49i)10-s + 2.08i·11-s + (1.89 − 5.69i)12-s + (8.39 − 8.39i)13-s + (9.39 + 3.12i)14-s + (11.4 + 9.65i)15-s − 4·16-s + (−4.96 + 4.96i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (−0.948 − 0.315i)3-s + 0.5i·4-s + (−0.929 − 0.369i)5-s + (0.316 + 0.632i)6-s + (−0.893 + 0.448i)7-s + (0.250 − 0.250i)8-s + (0.800 + 0.599i)9-s + (0.279 + 0.649i)10-s + 0.189i·11-s + (0.157 − 0.474i)12-s + (0.645 − 0.645i)13-s + (0.671 + 0.222i)14-s + (0.765 + 0.643i)15-s − 0.250·16-s + (−0.292 + 0.292i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00318i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.00318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.999 + 0.00318i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (83, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.999 + 0.00318i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.555085 - 0.000884042i\)
\(L(\frac12)\)  \(\approx\)  \(0.555085 - 0.000884042i\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
3 \( 1 + (2.84 + 0.947i)T \)
5 \( 1 + (4.64 + 1.84i)T \)
7 \( 1 + (6.25 - 3.13i)T \)
good11 \( 1 - 2.08iT - 121T^{2} \)
13 \( 1 + (-8.39 + 8.39i)T - 169iT^{2} \)
17 \( 1 + (4.96 - 4.96i)T - 289iT^{2} \)
19 \( 1 - 17.3T + 361T^{2} \)
23 \( 1 + (-3.08 + 3.08i)T - 529iT^{2} \)
29 \( 1 - 39.1T + 841T^{2} \)
31 \( 1 - 42.3iT - 961T^{2} \)
37 \( 1 + (-36.7 + 36.7i)T - 1.36e3iT^{2} \)
41 \( 1 + 15.5T + 1.68e3T^{2} \)
43 \( 1 + (-22.8 - 22.8i)T + 1.84e3iT^{2} \)
47 \( 1 + (33.4 - 33.4i)T - 2.20e3iT^{2} \)
53 \( 1 + (59.7 - 59.7i)T - 2.80e3iT^{2} \)
59 \( 1 - 48.9iT - 3.48e3T^{2} \)
61 \( 1 - 82.9iT - 3.72e3T^{2} \)
67 \( 1 + (54.8 - 54.8i)T - 4.48e3iT^{2} \)
71 \( 1 + 74.9iT - 5.04e3T^{2} \)
73 \( 1 + (-75.1 + 75.1i)T - 5.32e3iT^{2} \)
79 \( 1 - 3.61iT - 6.24e3T^{2} \)
83 \( 1 + (-103. - 103. i)T + 6.88e3iT^{2} \)
89 \( 1 - 24.4iT - 7.92e3T^{2} \)
97 \( 1 + (35.3 + 35.3i)T + 9.40e3iT^{2} \)
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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

−12.23889403958899457066711858020, −11.19198025147080103850723697024, −10.41663092607919217051964161283, −9.255451898972985550945223733391, −8.162971166166973102192996516777, −7.13128410665835795259979762002, −5.97874580666826453811297655958, −4.58893239848682591740747741196, −3.13273064654896192590930765370, −0.956827764532975595716110492534, 0.58428109432823004782226359944, 3.52135284299472372318837591633, 4.73065489478060385149486988816, 6.26827457356484860430091621386, 6.84939189887238332287474517654, 7.971747839709693924794330707206, 9.347942966124480940036525664901, 10.16530949581228802705044947909, 11.19894570554021464861477755803, 11.76916248390803265124424664577

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