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Properties

Label 2-1950-65.4-c1-0-9
Degree $2$
Conductor $1950$
Sign $0.660 - 0.751i$
Analytic cond. $15.5708$
Root an. cond. $3.94598$
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 + (−0.866 − 0.5i)3-s + (−0.499 − 0.866i)4-s + (−0.866 + 0.499i)6-s + (2.29 + 3.96i)7-s − 0.999·8-s + (0.499 + 0.866i)9-s + (−0.563 − 0.325i)11-s + 0.999i·12-s + (3.35 − 1.31i)13-s + 4.58·14-s + (−0.5 + 0.866i)16-s + (−3.03 + 1.75i)17-s + 0.999·18-s + (−2.50 + 1.44i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.499 − 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.353 + 0.204i)6-s + (0.865 + 1.49i)7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (−0.169 − 0.0981i)11-s + 0.288i·12-s + (0.931 − 0.364i)13-s + 1.22·14-s + (−0.125 + 0.216i)16-s + (−0.735 + 0.424i)17-s + 0.235·18-s + (−0.575 + 0.332i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.660 - 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.660 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.660 - 0.751i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ 0.660 - 0.751i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.364579860\)
\(L(\frac12)\) \(\approx\) \(1.364579860\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 \)
13 \( 1 + (-3.35 + 1.31i)T \)
good7 \( 1 + (-2.29 - 3.96i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.563 + 0.325i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.03 - 1.75i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.50 - 1.44i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.46 + 1.42i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.78 - 8.28i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.58iT - 31T^{2} \)
37 \( 1 + (2.31 - 4.00i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.03 + 1.75i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.68 + 3.85i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.08T + 47T^{2} \)
53 \( 1 - 5.58iT - 53T^{2} \)
59 \( 1 + (6.47 - 3.73i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.31 - 5.74i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.58 + 7.93i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (11.2 - 6.51i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 1.04T + 73T^{2} \)
79 \( 1 - 3.46T + 79T^{2} \)
83 \( 1 - 10.5T + 83T^{2} \)
89 \( 1 + (-7.10 - 4.09i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.21 - 12.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.048434447316956484249412964730, −8.706656625190624522321910950787, −7.915745472793868948808384084202, −6.66998376564950935032826489146, −5.86368591470610369296501767895, −5.34063745866447665164590643619, −4.49126235504629659503089122697, −3.34779367597148480091348641465, −2.21059604143135426105156000944, −1.46520464422665936989238952354, 0.46986279896159769083489531648, 1.98308365434945552858353613800, 3.72213485575913234277576759748, 4.24890010572519636875882409964, 4.90130682259155176711200910223, 5.97612653868105233521643501103, 6.62793592474516422630279523343, 7.52163552610976330703801029198, 8.010283152513558848761360335716, 9.027375154603929166169738387792

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