Properties

Label 2-1950-65.64-c1-0-39
Degree $2$
Conductor $1950$
Sign $-0.124 + 0.992i$
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  + 2-s i·3-s + 4-s i·6-s + 8-s − 9-s − 4i·11-s i·12-s + (2 − 3i)13-s + 16-s − 4i·17-s − 18-s + 4i·19-s − 4i·22-s + 6i·23-s i·24-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577i·3-s + 0.5·4-s − 0.408i·6-s + 0.353·8-s − 0.333·9-s − 1.20i·11-s − 0.288i·12-s + (0.554 − 0.832i)13-s + 0.250·16-s − 0.970i·17-s − 0.235·18-s + 0.917i·19-s − 0.852i·22-s + 1.25i·23-s − 0.204i·24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.124 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.124 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.124 + 0.992i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ -0.124 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.522449936\)
\(L(\frac12)\) \(\approx\) \(2.522449936\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + iT \)
5 \( 1 \)
13 \( 1 + (-2 + 3i)T \)
good7 \( 1 + 7T^{2} \)
11 \( 1 + 4iT - 11T^{2} \)
17 \( 1 + 4iT - 17T^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + 10iT - 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 - 2iT - 41T^{2} \)
43 \( 1 + 12iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 2iT - 53T^{2} \)
59 \( 1 - 4iT - 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 - 10T + 67T^{2} \)
71 \( 1 + 12iT - 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 10iT - 89T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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

−8.886523840401281914411206428382, −7.84954046133758381871875453885, −7.57890789636835864850367380507, −6.32348519188408125550787316452, −5.82462603755263349736329311202, −5.14582123041252896236355918626, −3.76623472439731720679553981675, −3.21226331271031054578421884867, −2.04031592071084337002022513943, −0.71478162549208498852725680244, 1.61752912988400627581928835745, 2.69536341333025241394149460749, 3.79297305775890104753365929140, 4.53127134664982239360982595946, 5.08255065135544625293200762185, 6.32264056210951013726041810822, 6.73097409306985938747973742310, 7.78903191326942455805947105579, 8.696552147254585597012943373384, 9.434667563293574858454104998240

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