Properties

Label 2-1950-65.9-c1-0-36
Degree $2$
Conductor $1950$
Sign $-0.435 + 0.900i$
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 + (−1.73 + i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (2.5 − 4.33i)11-s − 0.999i·12-s + (−3.46 + i)13-s − 1.99·14-s + (−0.5 + 0.866i)16-s + (−1.73 + i)17-s + 0.999i·18-s + (−1 − 1.73i)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 + (−0.654 + 0.377i)7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (0.753 − 1.30i)11-s − 0.288i·12-s + (−0.960 + 0.277i)13-s − 0.534·14-s + (−0.125 + 0.216i)16-s + (−0.420 + 0.242i)17-s + 0.235i·18-s + (−0.229 − 0.397i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.435 + 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.435 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.435 + 0.900i$
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: \((2,\ 1950,\ (\ :1/2),\ -0.435 + 0.900i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7049284391\)
\(L(\frac12)\) \(\approx\) \(0.7049284391\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.46 - i)T \)
good7 \( 1 + (1.73 - i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.5 + 4.33i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.73 - i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.866 + 0.5i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.5 + 4.33i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 11T + 31T^{2} \)
37 \( 1 + (2.59 + 1.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1 + 1.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-9.52 + 5.5i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 9iT - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + (7.5 + 12.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5 + 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (13.8 + 8i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 - 11T + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 + (-1 + 1.73i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.73 - i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.018377814117975807433746457580, −7.971477712833679421797284325377, −7.15902617652239968563065554869, −6.32800109421215922739135478849, −5.94449468473898000115201957124, −4.99424847125880081981777695669, −4.03454564218917444515837502544, −3.11107804089958557859208640472, −2.00283536386392535551068372740, −0.20962138138586612378228140035, 1.51260533180443386977805239735, 2.70326902081056409607779985780, 3.83444926623896607947559928947, 4.46293730273541636729741416599, 5.28457297368841068058511503464, 6.18384023069064505850465277195, 7.04597521530385100724488584323, 7.46061015395356919352567465008, 9.061417411483040065867642892961, 9.525064258538017061396171610931

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