Properties

Label 2-1950-65.49-c1-0-30
Degree $2$
Conductor $1950$
Sign $-0.996 + 0.0776i$
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 + (−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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0776i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.996 + 0.0776i$
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: \((2,\ 1950,\ (\ :1/2),\ -0.996 + 0.0776i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3692146797\)
\(L(\frac12)\) \(\approx\) \(0.3692146797\)
\(L(\frac{3}{2})\) 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 \( 1 + (7.21 - 12.5i)T + (-48.5 - 84.0i)T^{2} \)
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

−8.906483940558432952314828876471, −8.180940894858319785005233481928, −7.49372122911473793392329918692, −6.40386249882381934630755645106, −5.68231380194977135980534709494, −4.61382512956884284291249965790, −3.37646668912167881161226250527, −2.66319147493757764564942638787, −1.96386332955484509218552224669, −0.13870271607291597194643740250, 1.37722934379442543461538216658, 2.97847724938441786722201405050, 3.79474575419592992252279820739, 4.69988177652612543698352353850, 5.58134806846802711416832619004, 6.84251684489559602893606055056, 7.15073809719169853341426010700, 7.85514777174456631532685033025, 8.878072131534690514249358187571, 9.482632250871599528466865055640

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