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Properties

Label 2-1050-7.4-c1-0-17
Degree $2$
Conductor $1050$
Sign $0.991 - 0.126i$
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  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s − 0.999·6-s + (2 − 1.73i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (2.5 − 4.33i)11-s + (−0.499 − 0.866i)12-s + 13-s + (2.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (1 − 1.73i)17-s + (0.499 − 0.866i)18-s + (−3.5 − 6.06i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s − 0.408·6-s + (0.755 − 0.654i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.753 − 1.30i)11-s + (−0.144 − 0.249i)12-s + 0.277·13-s + (0.668 + 0.231i)14-s + (−0.125 − 0.216i)16-s + (0.242 − 0.420i)17-s + (0.117 − 0.204i)18-s + (−0.802 − 1.39i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.991 - 0.126i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ 0.991 - 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.788109472\)
\(L(\frac12)\) \(\approx\) \(1.788109472\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 \)
7 \( 1 + (-2 + 1.73i)T \)
good11 \( 1 + (-2.5 + 4.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - T + 13T^{2} \)
17 \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (-3 + 5.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.5 - 4.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 9T + 41T^{2} \)
43 \( 1 - 10T + 43T^{2} \)
47 \( 1 + (-6.5 - 11.2i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3 - 5.19i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 + (2 - 3.46i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7 - 12.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10T + 83T^{2} \)
89 \( 1 + (5 + 8.66i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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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

−9.899765909449641144356693723034, −8.872646549810631368164433471788, −8.362018409722813014355108036590, −7.32161497568378477150502503111, −6.41979782070643375466106831190, −5.70791873969953932658680505403, −4.58609756827196261315479236059, −4.07461360718282739088559403278, −2.84294318080270307053696080587, −0.814867990252770269833364469561, 1.51872408554304091446836241848, 2.13770032939468137589282869593, 3.70338451388937210747419161297, 4.60159893461207423913874254458, 5.57892636235429746558871784286, 6.33266396547368135898643772412, 7.38899760279329238110308322989, 8.300346981177860970219670897280, 9.100267704336247307921981004885, 10.10707055583706186173978654607

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