↑

Properties

Label 2-1980-11.4-c1-0-15
Degree $2$
Conductor $1980$
Sign $0.368 + 0.929i$
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 + (−1.43 − 4.42i)7-s + (3.31 − 0.130i)11-s + (4.88 + 3.54i)13-s + (−2.68 + 1.95i)17-s + (0.846 − 2.60i)19-s + 1.97·23-s + (0.309 − 0.951i)25-s + (−0.708 − 2.18i)29-s + (5.62 + 4.08i)31-s + (−3.76 − 2.73i)35-s + (−1.18 − 3.64i)37-s + (−1.17 + 3.60i)41-s + 7.98·43-s + (3.75 − 11.5i)47-s + ⋯
L(s)  = 1  + (0.361 − 0.262i)5-s + (−0.543 − 1.67i)7-s + (0.999 − 0.0394i)11-s + (1.35 + 0.983i)13-s + (−0.651 + 0.473i)17-s + (0.194 − 0.597i)19-s + 0.412·23-s + (0.0618 − 0.190i)25-s + (−0.131 − 0.405i)29-s + (1.01 + 0.733i)31-s + (−0.636 − 0.462i)35-s + (−0.194 − 0.599i)37-s + (−0.183 + 0.563i)41-s + 1.21·43-s + (0.548 − 1.68i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.368 + 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.368 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1980\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $0.368 + 0.929i$
Analytic conductor: \(15.8103\)
Root analytic conductor: \(3.97622\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1980} (1621, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1980,\ (\ :1/2),\ 0.368 + 0.929i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.920354398\)
\(L(\frac12)\) \(\approx\) \(1.920354398\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.31 + 0.130i)T \)
good7 \( 1 + (1.43 + 4.42i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-4.88 - 3.54i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (2.68 - 1.95i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.846 + 2.60i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 1.97T + 23T^{2} \)
29 \( 1 + (0.708 + 2.18i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-5.62 - 4.08i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.18 + 3.64i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.17 - 3.60i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 7.98T + 43T^{2} \)
47 \( 1 + (-3.75 + 11.5i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (9.30 + 6.76i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (2.19 + 6.75i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.08 + 0.786i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 7.18T + 67T^{2} \)
71 \( 1 + (-5.45 + 3.96i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (2.69 + 8.28i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (4.43 + 3.22i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (6.64 - 4.82i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 11.4T + 89T^{2} \)
97 \( 1 + (5.29 + 3.84i)T + (29.9 + 92.2i)T^{2} \)
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

−9.079592527400753397830786325183, −8.383856437916450938553525924124, −7.24832513179865932058272295324, −6.59829708535371268654514654441, −6.18003166616389193500742121330, −4.72128048908885296644183083742, −4.04749135745352740577637967871, −3.38331497767547525080883879919, −1.74576808800828612881839586953, −0.792027903552969779912189727384, 1.29719676967308473823953061538, 2.59716308339777069058354011622, 3.24578156548487356066992110021, 4.40848819090582618344637266556, 5.70980976175049792603059982798, 5.96787219275945031667632336741, 6.73015181403212748000192051519, 7.88323424842573589382161351609, 8.801595857428060047261774357921, 9.133013359074002678759602681205

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