↑

Properties

Label 2-1950-65.49-c1-0-16
Degree $2$
Conductor $1950$
Sign $0.950 + 0.310i$
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

Related objects

Downloads

Learn more

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 + (0.366 − 0.633i)7-s + 0.999·8-s + (0.499 − 0.866i)9-s + (−4.09 + 2.36i)11-s + 0.999i·12-s + (−2.5 + 2.59i)13-s − 0.732·14-s + (−0.5 − 0.866i)16-s + (1.96 + 1.13i)17-s − 0.999·18-s + (1.09 + 0.633i)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 + (0.138 − 0.239i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (−1.23 + 0.713i)11-s + 0.288i·12-s + (−0.693 + 0.720i)13-s − 0.195·14-s + (−0.125 − 0.216i)16-s + (0.476 + 0.275i)17-s − 0.235·18-s + (0.251 + 0.145i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.310i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.950 + 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.950 + 0.310i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ 0.950 + 0.310i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.531033143\)
\(L(\frac12)\) \(\approx\) \(1.531033143\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 - 2.59i)T \)
good7 \( 1 + (-0.366 + 0.633i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (4.09 - 2.36i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.96 - 1.13i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.09 - 0.633i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.36 + 3.09i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.23 - 2.13i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.46iT - 31T^{2} \)
37 \( 1 + (-5.23 - 9.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-9.86 + 5.69i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.63 - 3.83i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 8.19T + 47T^{2} \)
53 \( 1 - 0.464iT - 53T^{2} \)
59 \( 1 + (6.92 + 4i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.598 - 1.03i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.56 - 9.63i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.09 + 0.633i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 9.73T + 73T^{2} \)
79 \( 1 - 9.46T + 79T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 + (2.19 - 1.26i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3 + 5.19i)T + (-48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.266162104662247928690438810806, −8.403125668359802578357947687410, −7.52673348538646354278427540788, −7.27433774901175210803273146438, −5.98890025400125078948367871776, −4.82279235618497936441431889487, −4.18257211885833806088839208388, −2.85580381149248382211547205684, −2.32257314621944695857693257046, −1.01860698723468507585142813022, 0.73113531555575922734967081888, 2.47011555471893639618725310927, 3.18451992999944285344997090915, 4.50446652111217597549946894911, 5.40182182218295441464243685702, 5.81582228857043371436225142245, 7.28502724448759995771455678235, 7.60984707284275095391136870674, 8.400347738381861814004683346258, 9.155440612618433380593529679429

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