Properties

Degree $2$
Conductor $1050$
Sign $0.441 - 0.897i$
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 + (−0.866 + 2.5i)7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (−3 + 5.19i)11-s + (0.866 − 0.499i)12-s + 4i·13-s + (0.500 + 2.59i)14-s + (−0.5 − 0.866i)16-s + (2.59 + 1.5i)17-s + (0.866 + 0.499i)18-s + (−2 − 3.46i)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 + (−0.327 + 0.944i)7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.904 + 1.56i)11-s + (0.249 − 0.144i)12-s + 1.10i·13-s + (0.133 + 0.694i)14-s + (−0.125 − 0.216i)16-s + (0.630 + 0.363i)17-s + (0.204 + 0.117i)18-s + (−0.458 − 0.794i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.441 - 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.441 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.441 - 0.897i$
Motivic weight: \(1\)
Character: $\chi_{1050} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ 0.441 - 0.897i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.282919946\)
\(L(\frac12)\) \(\approx\) \(2.282919946\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 \)
7 \( 1 + (0.866 - 2.5i)T \)
good11 \( 1 + (3 - 5.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 + (-2.59 - 1.5i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2 + 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.59 - 1.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.92 + 4i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 + (-7.79 + 4.5i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-10.3 - 6i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.92 + 4i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 9T + 71T^{2} \)
73 \( 1 + (-12.1 - 7i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.5 + 6.06i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 + (-1.5 - 2.59i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 17iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.06783087852723639595273511492, −9.345499590436660810476807158840, −8.602239988955811157661525439501, −7.46088642320246444504857177966, −6.67495449620542958797644509197, −5.54145640256597988457700014120, −4.74490855693351358488029120181, −3.88629644280544625628283227217, −2.60013347672798769382334623438, −2.00394974214287043147633418900, 0.77563940836790596139122098489, 2.74456253382226711071637230133, 3.37970273041077530920096457158, 4.40322265965327691128932082319, 5.68359393025157227187238677789, 6.19804490651383943895751881021, 7.40921036504126359539888414805, 7.988172662892135423169909264164, 8.552923917569834415450490119327, 9.964036705942057028807875038888

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