Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.411 + 0.911i$
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 + (−1.94 − 2.28i)3-s + 2i·4-s + (4.58 − 1.99i)5-s + (0.347 − 4.22i)6-s + (−4.23 − 5.57i)7-s + (−2 + 2i)8-s + (−1.46 + 8.87i)9-s + (6.58 + 2.58i)10-s − 14.6i·11-s + (4.57 − 3.88i)12-s + (3.48 − 3.48i)13-s + (1.34 − 9.80i)14-s + (−13.4 − 6.61i)15-s − 4·16-s + (20.1 − 20.1i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (−0.646 − 0.762i)3-s + 0.5i·4-s + (0.916 − 0.399i)5-s + (0.0579 − 0.704i)6-s + (−0.604 − 0.796i)7-s + (−0.250 + 0.250i)8-s + (−0.163 + 0.986i)9-s + (0.658 + 0.258i)10-s − 1.32i·11-s + (0.381 − 0.323i)12-s + (0.267 − 0.267i)13-s + (0.0961 − 0.700i)14-s + (−0.897 − 0.441i)15-s − 0.250·16-s + (1.18 − 1.18i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.411 + 0.911i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.411 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.411 + 0.911i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (83, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.411 + 0.911i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.35385 - 0.874285i\)
\(L(\frac12)\)  \(\approx\)  \(1.35385 - 0.874285i\)
\(L(2)\)   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 \( 1 + (1.94 + 2.28i)T \)
5 \( 1 + (-4.58 + 1.99i)T \)
7 \( 1 + (4.23 + 5.57i)T \)
good11 \( 1 + 14.6iT - 121T^{2} \)
13 \( 1 + (-3.48 + 3.48i)T - 169iT^{2} \)
17 \( 1 + (-20.1 + 20.1i)T - 289iT^{2} \)
19 \( 1 + 26.4T + 361T^{2} \)
23 \( 1 + (-2.68 + 2.68i)T - 529iT^{2} \)
29 \( 1 - 28.5T + 841T^{2} \)
31 \( 1 - 15.6iT - 961T^{2} \)
37 \( 1 + (-7.69 + 7.69i)T - 1.36e3iT^{2} \)
41 \( 1 + 37.9T + 1.68e3T^{2} \)
43 \( 1 + (-41.7 - 41.7i)T + 1.84e3iT^{2} \)
47 \( 1 + (21.0 - 21.0i)T - 2.20e3iT^{2} \)
53 \( 1 + (47.4 - 47.4i)T - 2.80e3iT^{2} \)
59 \( 1 - 61.6iT - 3.48e3T^{2} \)
61 \( 1 - 54.1iT - 3.72e3T^{2} \)
67 \( 1 + (-68.9 + 68.9i)T - 4.48e3iT^{2} \)
71 \( 1 + 65.9iT - 5.04e3T^{2} \)
73 \( 1 + (6.51 - 6.51i)T - 5.32e3iT^{2} \)
79 \( 1 - 42.7iT - 6.24e3T^{2} \)
83 \( 1 + (9.52 + 9.52i)T + 6.88e3iT^{2} \)
89 \( 1 + 19.3iT - 7.92e3T^{2} \)
97 \( 1 + (-84.6 - 84.6i)T + 9.40e3iT^{2} \)
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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

−12.32163191896199096228033805415, −11.07326030054244688439102923468, −10.17670684547505022289251536952, −8.778097239391300553246663518755, −7.69321706480487449472905080446, −6.48414573475529633490574611095, −5.93282116868229097882691667543, −4.78821149334363491023525098358, −2.98087177759059170112243004585, −0.862105086905449307468963163199, 2.05323603106399761578537988229, 3.57409390309597827888446891131, 4.90463472810663586199251331345, 5.96756489661317196538279750469, 6.63307536017597901209272607520, 8.756447863533085723656142631424, 9.982299861991150681467796120393, 10.11233585966696093273203735428, 11.33610642713000888604166567057, 12.47110099196454547828510804020

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