Properties

Label 2-1950-65.9-c1-0-30
Degree $2$
Conductor $1950$
Sign $0.848 - 0.529i$
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 + (1.5 − 2.59i)11-s + 0.999i·12-s + (−3.46 − i)13-s + 1.99·14-s + (−0.5 + 0.866i)16-s + (5.19 − 3i)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 + (0.452 − 0.783i)11-s + 0.288i·12-s + (−0.960 − 0.277i)13-s + 0.534·14-s + (−0.125 + 0.216i)16-s + (1.26 − 0.727i)17-s + 0.235i·18-s + (0.229 + 0.397i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.848 - 0.529i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.848 - 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.848 - 0.529i$
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: \((2,\ 1950,\ (\ :1/2),\ 0.848 - 0.529i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.411006531\)
\(L(\frac12)\) \(\approx\) \(3.411006531\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-5.19 + 3i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.59 - 1.5i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5T + 31T^{2} \)
37 \( 1 + (-6.06 - 3.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (3 - 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.866 + 0.5i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 3iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + (4.5 + 7.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.92 + 4i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6 - 10.3i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 14iT - 73T^{2} \)
79 \( 1 + 5T + 79T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 + (9 - 15.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-12.1 + 7i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.299274077842347341046187101022, −8.070400040234568013526882291418, −7.895263227896869717851341799838, −6.95228655045403814499448662863, −5.96201138433748158462934882815, −5.08170376638841781803314581391, −4.45917400802815873378114971710, −3.38820413033894301381210582907, −2.72295003528674099123763671936, −1.18664031949751757219996932717, 1.27682277937301044327760350795, 2.22693088884867753732424398697, 3.10319032084578666643289212200, 4.22611763700011630604537169879, 4.91371659556913037704406045109, 5.79580795665875483145143530965, 6.82945604391956152344079175152, 7.47549909894048174784743808608, 8.305789170997734472666605028128, 9.190929230859424639742164182990

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