Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $-0.873 - 0.486i$
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 + (−1.86 + 2.35i)3-s + 2i·4-s + (1.91 + 4.61i)5-s + (−4.21 + 0.488i)6-s + (6.26 + 3.12i)7-s + (−2 + 2i)8-s + (−2.05 − 8.76i)9-s + (−2.70 + 6.53i)10-s + 0.117i·11-s + (−4.70 − 3.72i)12-s + (−9.72 + 9.72i)13-s + (3.13 + 9.39i)14-s + (−14.4 − 4.09i)15-s − 4·16-s + (13.8 − 13.8i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (−0.620 + 0.783i)3-s + 0.5i·4-s + (0.383 + 0.923i)5-s + (−0.702 + 0.0814i)6-s + (0.894 + 0.446i)7-s + (−0.250 + 0.250i)8-s + (−0.228 − 0.973i)9-s + (−0.270 + 0.653i)10-s + 0.0106i·11-s + (−0.391 − 0.310i)12-s + (−0.747 + 0.747i)13-s + (0.223 + 0.670i)14-s + (−0.961 − 0.273i)15-s − 0.250·16-s + (0.815 − 0.815i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.873 - 0.486i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.873 - 0.486i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.873 - 0.486i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (83, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ -0.873 - 0.486i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.419016 + 1.61390i\)
\(L(\frac12)\)  \(\approx\)  \(0.419016 + 1.61390i\)
\(L(2)\)   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 \( 1 + (1.86 - 2.35i)T \)
5 \( 1 + (-1.91 - 4.61i)T \)
7 \( 1 + (-6.26 - 3.12i)T \)
good11 \( 1 - 0.117iT - 121T^{2} \)
13 \( 1 + (9.72 - 9.72i)T - 169iT^{2} \)
17 \( 1 + (-13.8 + 13.8i)T - 289iT^{2} \)
19 \( 1 + 29.7T + 361T^{2} \)
23 \( 1 + (2.25 - 2.25i)T - 529iT^{2} \)
29 \( 1 - 46.0T + 841T^{2} \)
31 \( 1 + 1.50iT - 961T^{2} \)
37 \( 1 + (-5.32 + 5.32i)T - 1.36e3iT^{2} \)
41 \( 1 + 13.4T + 1.68e3T^{2} \)
43 \( 1 + (-36.8 - 36.8i)T + 1.84e3iT^{2} \)
47 \( 1 + (-29.7 + 29.7i)T - 2.20e3iT^{2} \)
53 \( 1 + (-59.8 + 59.8i)T - 2.80e3iT^{2} \)
59 \( 1 - 84.9iT - 3.48e3T^{2} \)
61 \( 1 + 34.8iT - 3.72e3T^{2} \)
67 \( 1 + (34.4 - 34.4i)T - 4.48e3iT^{2} \)
71 \( 1 - 77.6iT - 5.04e3T^{2} \)
73 \( 1 + (-41.3 + 41.3i)T - 5.32e3iT^{2} \)
79 \( 1 - 0.865iT - 6.24e3T^{2} \)
83 \( 1 + (-99.0 - 99.0i)T + 6.88e3iT^{2} \)
89 \( 1 + 129. iT - 7.92e3T^{2} \)
97 \( 1 + (15.7 + 15.7i)T + 9.40e3iT^{2} \)
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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

−12.28974775999265749782017598598, −11.63771445639781537556629213037, −10.69451926682076469861449640478, −9.748855975611572126127277487056, −8.578393996319277690179501256210, −7.18755012112163832965533662965, −6.21736825555791917690033082025, −5.19538818377335946575214236323, −4.20797962246606298799174097360, −2.57709143269618267068702354630, 0.897010229355952556639695721454, 2.18049924974975951475462256203, 4.39139090420537416487442639210, 5.27443557403322426359563709147, 6.26849267134547143251367956592, 7.73023807749212907113431988487, 8.578261826932892363488212874568, 10.23816284431982839194977453745, 10.79136567687928690001968974321, 12.21106375195387321185407276234

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