Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.758 + 0.651i$
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 + (−2.28 − 1.94i)3-s + 2i·4-s + (4.58 − 1.99i)5-s + (0.347 + 4.22i)6-s + (5.57 + 4.23i)7-s + (2 − 2i)8-s + (1.46 + 8.87i)9-s + (−6.58 − 2.58i)10-s + 14.6i·11-s + (3.88 − 4.57i)12-s + (−3.48 + 3.48i)13-s + (−1.34 − 9.80i)14-s + (−14.3 − 4.32i)15-s − 4·16-s + (20.1 − 20.1i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (−0.762 − 0.646i)3-s + 0.5i·4-s + (0.916 − 0.399i)5-s + (0.0579 + 0.704i)6-s + (0.796 + 0.604i)7-s + (0.250 − 0.250i)8-s + (0.163 + 0.986i)9-s + (−0.658 − 0.258i)10-s + 1.32i·11-s + (0.323 − 0.381i)12-s + (−0.267 + 0.267i)13-s + (−0.0961 − 0.700i)14-s + (−0.957 − 0.288i)15-s − 0.250·16-s + (1.18 − 1.18i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.758 + 0.651i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.758 + 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.758 + 0.651i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (83, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.758 + 0.651i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.12788 - 0.418125i\)
\(L(\frac12)\)  \(\approx\)  \(1.12788 - 0.418125i\)
\(L(2)\)   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 \( 1 + (2.28 + 1.94i)T \)
5 \( 1 + (-4.58 + 1.99i)T \)
7 \( 1 + (-5.57 - 4.23i)T \)
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

−11.99500432641393137413933833809, −11.28691118303670832544563402078, −9.907498099915475979544312666862, −9.419797412214707978346051649150, −7.909749294959012797161173845968, −7.12942985548933972231771626230, −5.58022730241305875763729016864, −4.85817532045905542630718240975, −2.36887319920942557014381728669, −1.29344790169983435542084387050, 1.14195176532973354055453107320, 3.55733235894958434362751342784, 5.33032932984853710369213303163, 5.80819690098104184367731829677, 7.08826534212976577331152816283, 8.279696199884010117000220216673, 9.482387043907186201661922029410, 10.37866537129761438602242419735, 10.88904649601049128755281724993, 11.94679955919377379254740708486

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