Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.557 - 0.830i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.36 + 0.366i)2-s + (−1.66 − 2.49i)3-s + (1.73 + i)4-s + (1.09 + 4.87i)5-s + (−1.36 − 4.01i)6-s + (−1.72 + 6.78i)7-s + (1.99 + 2i)8-s + (−3.43 + 8.31i)9-s + (−0.293 + 7.06i)10-s + (5.25 + 3.03i)11-s + (−0.394 − 5.98i)12-s + (4.95 + 4.95i)13-s + (−4.83 + 8.63i)14-s + (10.3 − 10.8i)15-s + (1.99 + 3.46i)16-s + (24.7 − 6.62i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (−0.555 − 0.831i)3-s + (0.433 + 0.250i)4-s + (0.218 + 0.975i)5-s + (−0.227 − 0.669i)6-s + (−0.246 + 0.969i)7-s + (0.249 + 0.250i)8-s + (−0.381 + 0.924i)9-s + (−0.0293 + 0.706i)10-s + (0.477 + 0.275i)11-s + (−0.0328 − 0.498i)12-s + (0.381 + 0.381i)13-s + (−0.345 + 0.616i)14-s + (0.689 − 0.724i)15-s + (0.124 + 0.216i)16-s + (1.45 − 0.389i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.557 - 0.830i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.557 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.557 - 0.830i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (17, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.557 - 0.830i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.68026 + 0.895693i\)
\(L(\frac12)\)  \(\approx\)  \(1.68026 + 0.895693i\)
\(L(2)\)   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 \( 1 + (-1.36 - 0.366i)T \)
3 \( 1 + (1.66 + 2.49i)T \)
5 \( 1 + (-1.09 - 4.87i)T \)
7 \( 1 + (1.72 - 6.78i)T \)
good11 \( 1 + (-5.25 - 3.03i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-4.95 - 4.95i)T + 169iT^{2} \)
17 \( 1 + (-24.7 + 6.62i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (8.47 + 14.6i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (7.44 - 27.7i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 - 29.7T + 841T^{2} \)
31 \( 1 + (36.5 + 21.1i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (12.4 - 46.2i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 2.99T + 1.68e3T^{2} \)
43 \( 1 + (-22.2 + 22.2i)T - 1.84e3iT^{2} \)
47 \( 1 + (-16.3 + 61.1i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-1.70 + 0.456i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (72.6 + 41.9i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (18.0 - 10.4i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-32.1 + 8.61i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 10.6iT - 5.04e3T^{2} \)
73 \( 1 + (-94.5 + 25.3i)T + (4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (-74.6 + 43.1i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-47.0 + 47.0i)T - 6.88e3iT^{2} \)
89 \( 1 + (-138. + 80.0i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (81.7 - 81.7i)T - 9.40e3iT^{2} \)
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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

−12.10159696634673086475120515193, −11.74166321673577384569212441181, −10.66994226782581281830300476660, −9.385018280178760049088899128291, −7.899170428918328057070957830810, −6.89121841134561568866645681563, −6.13210441777554733087537826083, −5.23179463195865922307252346132, −3.32485772666970293992205407020, −2.00562477443287170428804462285, 0.998832820137927859811652397147, 3.55433747312490104943030613899, 4.36673025054316244894242524374, 5.53767159609364471217887323261, 6.37702991462072272905549145628, 8.021402777839667878354631108042, 9.283923674397646560420470509944, 10.31052259718017102119393782317, 10.87803028690326942229871391701, 12.34731133426133965180656792368

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