Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.447 + 0.894i$
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 + (2.14 + 2.09i)3-s + 2i·4-s + (1.13 − 4.86i)5-s + (−0.0523 − 4.24i)6-s + (2.31 − 6.60i)7-s + (2 − 2i)8-s + (0.222 + 8.99i)9-s + (−6.00 + 3.73i)10-s − 16.9i·11-s + (−4.18 + 4.29i)12-s + (−10.2 + 10.2i)13-s + (−8.91 + 4.29i)14-s + (12.6 − 8.07i)15-s − 4·16-s + (8.79 − 8.79i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (0.715 + 0.698i)3-s + 0.5i·4-s + (0.227 − 0.973i)5-s + (−0.00872 − 0.707i)6-s + (0.330 − 0.943i)7-s + (0.250 − 0.250i)8-s + (0.0246 + 0.999i)9-s + (−0.600 + 0.373i)10-s − 1.54i·11-s + (−0.349 + 0.357i)12-s + (−0.786 + 0.786i)13-s + (−0.637 + 0.306i)14-s + (0.842 − 0.538i)15-s − 0.250·16-s + (0.517 − 0.517i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.447 + 0.894i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (83, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.447 + 0.894i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.32453 - 0.818758i\)
\(L(\frac12)\)  \(\approx\)  \(1.32453 - 0.818758i\)
\(L(2)\)   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 \( 1 + (-2.14 - 2.09i)T \)
5 \( 1 + (-1.13 + 4.86i)T \)
7 \( 1 + (-2.31 + 6.60i)T \)
good11 \( 1 + 16.9iT - 121T^{2} \)
13 \( 1 + (10.2 - 10.2i)T - 169iT^{2} \)
17 \( 1 + (-8.79 + 8.79i)T - 289iT^{2} \)
19 \( 1 - 24.7T + 361T^{2} \)
23 \( 1 + (-19.2 + 19.2i)T - 529iT^{2} \)
29 \( 1 - 1.67T + 841T^{2} \)
31 \( 1 - 36.8iT - 961T^{2} \)
37 \( 1 + (-40.5 + 40.5i)T - 1.36e3iT^{2} \)
41 \( 1 - 0.885T + 1.68e3T^{2} \)
43 \( 1 + (9.87 + 9.87i)T + 1.84e3iT^{2} \)
47 \( 1 + (33.7 - 33.7i)T - 2.20e3iT^{2} \)
53 \( 1 + (11.9 - 11.9i)T - 2.80e3iT^{2} \)
59 \( 1 - 50.5iT - 3.48e3T^{2} \)
61 \( 1 - 80.6iT - 3.72e3T^{2} \)
67 \( 1 + (4.46 - 4.46i)T - 4.48e3iT^{2} \)
71 \( 1 - 137. iT - 5.04e3T^{2} \)
73 \( 1 + (-53.3 + 53.3i)T - 5.32e3iT^{2} \)
79 \( 1 - 127. iT - 6.24e3T^{2} \)
83 \( 1 + (60.0 + 60.0i)T + 6.88e3iT^{2} \)
89 \( 1 + 51.3iT - 7.92e3T^{2} \)
97 \( 1 + (0.274 + 0.274i)T + 9.40e3iT^{2} \)
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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

−11.73672750376133553344426738237, −10.87129876068116904502640049489, −9.854584916746385444373125930627, −9.104253892265899389816262646015, −8.293512799266217346303782573187, −7.29859274531306661606589600071, −5.27989059807420103031360446031, −4.23179364922959627040213647500, −2.94651247651673935087375856898, −1.05261901623184565059869763912, 1.85618793745476090557090404556, 3.03764645007367156350724478923, 5.20712936177019833465806910702, 6.43023465789370610598359869194, 7.49714397288020969138102087341, 7.950026608270931968217451550704, 9.549811342589750564720280557698, 9.811217289553898998461876201369, 11.42025195907949893172378949667, 12.34245301824387648666733763742

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