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  :  $\chi_{210} (167, \cdot )$
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

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

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