↑

Properties

Label 2-1980-11.3-c1-0-6
Degree $2$
Conductor $1980$
Sign $0.679 - 0.733i$
Analytic cond. $15.8103$
Root an. cond. $3.97622$
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  + (0.809 + 0.587i)5-s + (0.177 − 0.546i)7-s + (−3.04 − 1.31i)11-s + (−0.806 + 0.585i)13-s + (−0.555 − 0.403i)17-s + (2.32 + 7.16i)19-s + 6.80·23-s + (0.309 + 0.951i)25-s + (0.0445 − 0.137i)29-s + (−1.73 + 1.26i)31-s + (0.464 − 0.337i)35-s + (−0.123 + 0.381i)37-s + (1.42 + 4.38i)41-s + 11.3·43-s + (2.23 + 6.88i)47-s + ⋯
L(s)  = 1  + (0.361 + 0.262i)5-s + (0.0671 − 0.206i)7-s + (−0.917 − 0.397i)11-s + (−0.223 + 0.162i)13-s + (−0.134 − 0.0977i)17-s + (0.534 + 1.64i)19-s + 1.41·23-s + (0.0618 + 0.190i)25-s + (0.00827 − 0.0254i)29-s + (−0.312 + 0.226i)31-s + (0.0785 − 0.0570i)35-s + (−0.0203 + 0.0626i)37-s + (0.222 + 0.684i)41-s + 1.72·43-s + (0.326 + 1.00i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.679 - 0.733i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.679 - 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1980\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $0.679 - 0.733i$
Analytic conductor: \(15.8103\)
Root analytic conductor: \(3.97622\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1980} (1081, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1980,\ (\ :1/2),\ 0.679 - 0.733i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.691036867\)
\(L(\frac12)\) \(\approx\) \(1.691036867\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (3.04 + 1.31i)T \)
good7 \( 1 + (-0.177 + 0.546i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (0.806 - 0.585i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.555 + 0.403i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.32 - 7.16i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 6.80T + 23T^{2} \)
29 \( 1 + (-0.0445 + 0.137i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.73 - 1.26i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.123 - 0.381i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.42 - 4.38i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 11.3T + 43T^{2} \)
47 \( 1 + (-2.23 - 6.88i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (4.67 - 3.39i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-4.39 + 13.5i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-8.65 - 6.28i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 5.48T + 67T^{2} \)
71 \( 1 + (4.16 + 3.02i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (4.36 - 13.4i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-5.93 + 4.31i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (9.19 + 6.68i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 9.42T + 89T^{2} \)
97 \( 1 + (4.93 - 3.58i)T + (29.9 - 92.2i)T^{2} \)
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

−9.373267930056708685657504759295, −8.440172685893114395171191704638, −7.66907172055986050400950830629, −7.02948112437444387900002086587, −5.95847241962114531269606620473, −5.40465231568885517028273551867, −4.39237668401033044036008303680, −3.30644757997114716662111476952, −2.46557998147660833976380473076, −1.14483904752627564287193777946, 0.69495762553599671942814596634, 2.22948847815796199475462310882, 2.93415664337152519746997362514, 4.28211186420262394740200177774, 5.17466447081221325513320697631, 5.61477248636350936210159415617, 6.94292635421185127743487120529, 7.32486299923900368141505017535, 8.420052159922344015464539991898, 9.088952667025080799884082306319

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