Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5^{2} \cdot 7 $
Sign $0.669 - 0.742i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s − 0.999·6-s + (−1.73 − 2i)7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (2.5 + 4.33i)11-s + (0.866 + 0.499i)12-s + 5i·13-s + (0.499 + 2.59i)14-s + (−0.5 + 0.866i)16-s + (−3.46 + 2i)17-s + (−0.866 + 0.499i)18-s + (−3.5 + 6.06i)19-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s − 0.408·6-s + (−0.654 − 0.755i)7-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (0.753 + 1.30i)11-s + (0.249 + 0.144i)12-s + 1.38i·13-s + (0.133 + 0.694i)14-s + (−0.125 + 0.216i)16-s + (−0.840 + 0.485i)17-s + (−0.204 + 0.117i)18-s + (−0.802 + 1.39i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 - 0.742i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.669 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.669 - 0.742i$
motivic weight  =  \(1\)
character  :  $\chi_{1050} (499, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1050,\ (\ :1/2),\ 0.669 - 0.742i)\)
\(L(1)\)  \(\approx\)  \(1.070040171\)
\(L(\frac12)\)  \(\approx\)  \(1.070040171\)
\(L(\frac{3}{2})\)   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 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 \)
7 \( 1 + (1.73 + 2i)T \)
good11 \( 1 + (-2.5 - 4.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 5iT - 13T^{2} \)
17 \( 1 + (3.46 - 2i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.5 - 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.866 + 0.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 5T + 41T^{2} \)
43 \( 1 + 12iT - 43T^{2} \)
47 \( 1 + (-9.52 - 5.5i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7.79 + 4.5i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2 - 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.3 - 6i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + (8.66 - 5i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (6 - 10.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 + (-7 + 12.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 8iT - 97T^{2} \)
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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

−10.01369004985183568090038323070, −9.133633559248558376613820815514, −8.653230897741031791154182827267, −7.37192007754666077860539818282, −6.97991061745274308466963357169, −6.15343457832562239082049460922, −4.17549894720672970958948644516, −3.97641672944950888662889964838, −2.34152392308899562247198033601, −1.47944023192445944701519406839, 0.56281755428133394351010228655, 2.49987477204592274667071763532, 3.22710904009703522802973089291, 4.60876521342614711698159819409, 5.78067556119195978892506353556, 6.38985338402122664232791814522, 7.40755629207756290983631942276, 8.469162812169749883588159781994, 8.906909091532316967633236372275, 9.487042605049337052401512966940

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