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Properties

Label 2-1050-35.3-c1-0-13
Degree $2$
Conductor $1050$
Sign $0.683 + 0.730i$
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 + (2.63 − 0.189i)7-s + (−0.707 − 0.707i)8-s + (−0.866 + 0.499i)9-s + (−0.5 + 0.866i)11-s + (0.258 − 0.965i)12-s + (1.60 − 1.60i)13-s + (−2.59 − 0.5i)14-s + (0.500 + 0.866i)16-s + (0.517 − 0.138i)17-s + (0.965 − 0.258i)18-s + (2.86 + 4.96i)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 + (0.997 − 0.0716i)7-s + (−0.249 − 0.249i)8-s + (−0.288 + 0.166i)9-s + (−0.150 + 0.261i)11-s + (0.0747 − 0.278i)12-s + (0.444 − 0.444i)13-s + (−0.694 − 0.133i)14-s + (0.125 + 0.216i)16-s + (0.125 − 0.0336i)17-s + (0.227 − 0.0610i)18-s + (0.657 + 1.13i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.683 + 0.730i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.683 + 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.683 + 0.730i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ 0.683 + 0.730i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.267146088\)
\(L(\frac12)\) \(\approx\) \(1.267146088\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.63 + 0.189i)T \)
good11 \( 1 + (0.5 - 0.866i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.60 + 1.60i)T - 13iT^{2} \)
17 \( 1 + (-0.517 + 0.138i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-2.86 - 4.96i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.05 + 7.65i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 6.92iT - 29T^{2} \)
31 \( 1 + (0.464 + 0.267i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-9.58 - 2.56i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 5.73iT - 41T^{2} \)
43 \( 1 + (3.48 + 3.48i)T + 43iT^{2} \)
47 \( 1 + (2.38 - 8.88i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.965 - 0.258i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-2.26 + 3.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.92 + 2.26i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.517 + 1.93i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 4.92T + 71T^{2} \)
73 \( 1 + (1.03 + 3.86i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-14.6 + 8.46i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.34 + 7.34i)T - 83iT^{2} \)
89 \( 1 + (3.46 + 6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.72 + 7.72i)T + 97iT^{2} \)
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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

−9.870220882861396875938657861673, −8.796495181493261517928250401832, −8.124143540190082998192243311663, −7.54632708501931173086022720871, −6.59952622303631242641170061603, −5.62567997559179871349852979877, −4.62526963491269380480544724844, −3.25055019405868756347445494129, −2.01580043639252985362491514988, −0.962550171260421023745527618192, 1.09847944289844968194855711037, 2.51989028844556403997757285253, 3.85041360758353400398175635062, 4.96506992706367675928202307584, 5.68102420315699528684690873620, 6.75838931316225566425894959424, 7.74266325038779764569086878182, 8.359313964560183609885525475194, 9.343608878067373854816972059532, 9.759023485879868265774747646591

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