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Properties

Label 2-1950-65.9-c1-0-8
Degree $2$
Conductor $1950$
Sign $-0.175 - 0.984i$
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  + (−0.866 − 0.5i)2-s + (0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + (−0.499 − 0.866i)6-s + (−0.486 + 0.280i)7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (−2.06 + 3.57i)11-s + 0.999i·12-s + (2.21 − 2.84i)13-s + 0.561·14-s + (−0.5 + 0.866i)16-s + (−2.70 + 1.56i)17-s − 0.999i·18-s + (0.280 + 0.486i)19-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.499 + 0.288i)3-s + (0.249 + 0.433i)4-s + (−0.204 − 0.353i)6-s + (−0.183 + 0.106i)7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.621 + 1.07i)11-s + 0.288i·12-s + (0.615 − 0.788i)13-s + 0.150·14-s + (−0.125 + 0.216i)16-s + (−0.655 + 0.378i)17-s − 0.235i·18-s + (0.0644 + 0.111i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.175 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.175 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.175 - 0.984i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (1699, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ -0.175 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.016927979\)
\(L(\frac12)\) \(\approx\) \(1.016927979\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 \)
13 \( 1 + (-2.21 + 2.84i)T \)
good7 \( 1 + (0.486 - 0.280i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.06 - 3.57i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.70 - 1.56i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.280 - 0.486i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.05 - 2.34i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.21 + 2.11i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.68T + 31T^{2} \)
37 \( 1 + (-3.57 - 2.06i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.12 - 10.6i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.379 - 0.219i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 7iT - 47T^{2} \)
53 \( 1 + 8.56iT - 53T^{2} \)
59 \( 1 + (-3.21 - 5.57i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3 + 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.94 - 1.12i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.56 - 11.3i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 9.36iT - 73T^{2} \)
79 \( 1 + 11.5T + 79T^{2} \)
83 \( 1 - 7.12iT - 83T^{2} \)
89 \( 1 + (9.40 - 16.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.16 - 3.56i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.614777887878502140980595784419, −8.607439306273497217253940385761, −8.035686104482595307855305098325, −7.29779560214908861527763393725, −6.40469168646019038381764816527, −5.28876894892621310261575103345, −4.36975576586746713546025010865, −3.34142827044178608614368269369, −2.54089095482206124678955399492, −1.42994971897234167969885681697, 0.42088625231916027229762144643, 1.80165861307163511130017118014, 2.89030819897034164576110448280, 3.85511980865007324380485656128, 5.08316099294939658340685390043, 5.96533219008411930210916195829, 6.85879501710999360135690887957, 7.32686020556415370366770029867, 8.479359770652502605454197236079, 8.736954638332719612805729821538

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