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Properties

Label 2-1050-35.12-c1-0-14
Degree $2$
Conductor $1050$
Sign $0.812 + 0.582i$
Analytic cond. $8.38429$
Root an. cond. $2.89556$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.965 − 0.258i)2-s + (0.258 − 0.965i)3-s + (0.866 − 0.499i)4-s − i·6-s + (2.64 − 0.153i)7-s + (0.707 − 0.707i)8-s + (−0.866 − 0.499i)9-s + (2.27 + 3.94i)11-s + (−0.258 − 0.965i)12-s + (1.77 + 1.77i)13-s + (2.51 − 0.831i)14-s + (0.500 − 0.866i)16-s + (3.98 + 1.06i)17-s + (−0.965 − 0.258i)18-s + (−1.88 + 3.27i)19-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.149 − 0.557i)3-s + (0.433 − 0.249i)4-s − 0.408i·6-s + (0.998 − 0.0579i)7-s + (0.249 − 0.249i)8-s + (−0.288 − 0.166i)9-s + (0.686 + 1.18i)11-s + (−0.0747 − 0.278i)12-s + (0.493 + 0.493i)13-s + (0.671 − 0.222i)14-s + (0.125 − 0.216i)16-s + (0.966 + 0.258i)17-s + (−0.227 − 0.0610i)18-s + (−0.433 + 0.750i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.812 + 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.812 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.812 + 0.582i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ 0.812 + 0.582i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.991354591\)
\(L(\frac12)\) \(\approx\) \(2.991354591\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
3 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 \)
7 \( 1 + (-2.64 + 0.153i)T \)
good11 \( 1 + (-2.27 - 3.94i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.77 - 1.77i)T + 13iT^{2} \)
17 \( 1 + (-3.98 - 1.06i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.88 - 3.27i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.08 + 7.77i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 1.55iT - 29T^{2} \)
31 \( 1 + (-3.37 + 1.94i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (11.0 - 2.95i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 11.3iT - 41T^{2} \)
43 \( 1 + (0.367 - 0.367i)T - 43iT^{2} \)
47 \( 1 + (1.30 + 4.87i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (8.14 + 2.18i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-0.221 - 0.383i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.09 - 4.09i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.41 - 8.99i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 6.68T + 71T^{2} \)
73 \( 1 + (-1.12 + 4.20i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-4.08 - 2.35i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.21 + 3.21i)T + 83iT^{2} \)
89 \( 1 + (3.02 - 5.23i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.462 + 0.462i)T - 97iT^{2} \)
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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

−10.09251193700476859824901141744, −8.816666089892466407341842699558, −8.144535812832115405772909399960, −7.16134491717770433070610625302, −6.49876308279162936036324448523, −5.47249062902824872681562972381, −4.47878288242061646220281144741, −3.71973641552967073631567610726, −2.18865362109972699556429753446, −1.45263513121864355692270095312, 1.41845302071805948534261963659, 3.04215889288583402830744942870, 3.74930715931906250385446824385, 4.83909698005433956816892671041, 5.56169740054855629714208358342, 6.36493730044988504298224135423, 7.62058446252961570668793851316, 8.274873469377541030403939437200, 9.052560272929101723453967693488, 10.07584515394297445875663306633

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