↑

Properties

Label 2-760-19.11-c1-0-5
Degree $2$
Conductor $760$
Sign $0.941 - 0.337i$
Analytic cond. $6.06863$
Root an. cond. $2.46345$
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  + (0.548 − 0.950i)3-s + (−0.5 + 0.866i)5-s − 0.416·7-s + (0.897 + 1.55i)9-s + 2.17·11-s + (2.13 + 3.70i)13-s + (0.548 + 0.950i)15-s + (0.602 − 1.04i)17-s + (−4.32 + 0.572i)19-s + (−0.228 + 0.395i)21-s + (0.224 + 0.389i)23-s + (−0.499 − 0.866i)25-s + 5.26·27-s + (2.94 + 5.09i)29-s + 8.26·31-s + ⋯
L(s)  = 1  + (0.316 − 0.548i)3-s + (−0.223 + 0.387i)5-s − 0.157·7-s + (0.299 + 0.518i)9-s + 0.655·11-s + (0.592 + 1.02i)13-s + (0.141 + 0.245i)15-s + (0.146 − 0.252i)17-s + (−0.991 + 0.131i)19-s + (−0.0498 + 0.0863i)21-s + (0.0468 + 0.0811i)23-s + (−0.0999 − 0.173i)25-s + 1.01·27-s + (0.546 + 0.946i)29-s + 1.48·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 - 0.337i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.941 - 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign: $0.941 - 0.337i$
Analytic conductor: \(6.06863\)
Root analytic conductor: \(2.46345\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{760} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 760,\ (\ :1/2),\ 0.941 - 0.337i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.68900 + 0.293936i\)
\(L(\frac12)\) \(\approx\) \(1.68900 + 0.293936i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (4.32 - 0.572i)T \)
good3 \( 1 + (-0.548 + 0.950i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + 0.416T + 7T^{2} \)
11 \( 1 - 2.17T + 11T^{2} \)
13 \( 1 + (-2.13 - 3.70i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.602 + 1.04i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-0.224 - 0.389i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.94 - 5.09i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 8.26T + 31T^{2} \)
37 \( 1 - 6.80T + 37T^{2} \)
41 \( 1 + (2.30 - 3.99i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.80 + 4.85i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.07 - 1.85i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.00 + 8.67i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.74 + 3.02i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.86 - 8.43i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.89 + 6.74i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.09 + 10.5i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.534 + 0.924i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.04 - 12.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 9.73T + 83T^{2} \)
89 \( 1 + (4.19 + 7.25i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.91 - 6.78i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.43059321420032216289836789248, −9.477891183116731645634245480618, −8.543205268973394262020350936092, −7.85312819141019432992125689926, −6.72235940253024038463566588570, −6.42761066835830494015182774157, −4.83090274794799988161283050893, −3.90108656151089728725805652452, −2.63509180059583115298022630848, −1.44373222818674297066029506109, 0.989251596735325955379014634779, 2.82914592383782865748777191270, 3.91859668231606565691893828015, 4.57582986030935619475670363591, 5.95283898725918077838233632703, 6.65218787921129003602869079975, 8.003389512024305489493311075916, 8.568918574639169565896423506487, 9.493001615587211546332865597903, 10.14415798720454728015880003551

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