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Properties

Label 2-1050-35.33-c1-0-9
Degree $2$
Conductor $1050$
Sign $0.815 - 0.578i$
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  + (−0.258 − 0.965i)2-s + (0.965 + 0.258i)3-s + (−0.866 + 0.499i)4-s − i·6-s + (2.46 + 0.965i)7-s + (0.707 + 0.707i)8-s + (0.866 + 0.499i)9-s + (1.92 + 3.33i)11-s + (−0.965 + 0.258i)12-s + (−4.42 + 4.42i)13-s + (0.295 − 2.62i)14-s + (0.500 − 0.866i)16-s + (0.364 − 1.36i)17-s + (0.258 − 0.965i)18-s + (−1.76 + 3.05i)19-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (0.557 + 0.149i)3-s + (−0.433 + 0.249i)4-s − 0.408i·6-s + (0.930 + 0.365i)7-s + (0.249 + 0.249i)8-s + (0.288 + 0.166i)9-s + (0.580 + 1.00i)11-s + (−0.278 + 0.0747i)12-s + (−1.22 + 1.22i)13-s + (0.0789 − 0.702i)14-s + (0.125 − 0.216i)16-s + (0.0884 − 0.330i)17-s + (0.0610 − 0.227i)18-s + (−0.404 + 0.700i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.815 - 0.578i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.815 - 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.815 - 0.578i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (943, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ 0.815 - 0.578i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.681907621\)
\(L(\frac12)\) \(\approx\) \(1.681907621\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
3 \( 1 + (-0.965 - 0.258i)T \)
5 \( 1 \)
7 \( 1 + (-2.46 - 0.965i)T \)
good11 \( 1 + (-1.92 - 3.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.42 - 4.42i)T - 13iT^{2} \)
17 \( 1 + (-0.364 + 1.36i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.76 - 3.05i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.36 - 0.364i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 6.66iT - 29T^{2} \)
31 \( 1 + (4.55 - 2.62i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.63 + 9.82i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 0.741iT - 41T^{2} \)
43 \( 1 + (-1.52 - 1.52i)T + 43iT^{2} \)
47 \( 1 + (-5.07 + 1.36i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.54 + 9.51i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-6.84 - 11.8i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.857 + 0.495i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.39 - 1.98i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 11.2T + 71T^{2} \)
73 \( 1 + (10.2 + 2.75i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-2.72 - 1.57i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.471 + 0.471i)T - 83iT^{2} \)
89 \( 1 + (-6.97 + 12.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.98 - 6.98i)T + 97iT^{2} \)
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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

−9.914693400611694300716386509082, −9.163156661965750535194444810423, −8.675192448533843597495691628697, −7.50962856671494548011047541358, −7.00739496147392071142437117577, −5.38062656401671321932419310738, −4.54425142972913331150711466504, −3.78963623940278224774906863587, −2.28032191773833646677613258852, −1.74710709027135068204618869364, 0.77141096120804784934901377554, 2.34440103601727250638752811432, 3.65599525061971288451957853728, 4.68827663134747186313738201917, 5.58584759187995192357666576516, 6.57882359261757980190257898498, 7.57358842347768242694549529823, 8.065550410129698582067738164654, 8.761279315897718380955704069477, 9.698530681860636765065396646143

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