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Properties

Label 2-1980-11.3-c1-0-1
Degree $2$
Conductor $1980$
Sign $-0.899 - 0.436i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.809 − 0.587i)5-s + (−1.24 + 3.84i)7-s + (0.375 − 3.29i)11-s + (0.139 − 0.101i)13-s + (0.864 + 0.627i)17-s + (0.140 + 0.432i)19-s + 0.873·23-s + (0.309 + 0.951i)25-s + (−1.10 + 3.41i)29-s + (−5.02 + 3.65i)31-s + (3.26 − 2.37i)35-s + (−0.440 + 1.35i)37-s + (−1.43 − 4.40i)41-s − 8.59·43-s + (3.75 + 11.5i)47-s + ⋯
L(s)  = 1  + (−0.361 − 0.262i)5-s + (−0.471 + 1.45i)7-s + (0.113 − 0.993i)11-s + (0.0386 − 0.0280i)13-s + (0.209 + 0.152i)17-s + (0.0322 + 0.0991i)19-s + 0.182·23-s + (0.0618 + 0.190i)25-s + (−0.205 + 0.633i)29-s + (−0.903 + 0.656i)31-s + (0.552 − 0.401i)35-s + (−0.0724 + 0.222i)37-s + (−0.223 − 0.688i)41-s − 1.31·43-s + (0.548 + 1.68i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.899 - 0.436i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.899 - 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1980\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $-0.899 - 0.436i$
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: \((2,\ 1980,\ (\ :1/2),\ -0.899 - 0.436i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5058599068\)
\(L(\frac12)\) \(\approx\) \(0.5058599068\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.375 + 3.29i)T \)
good7 \( 1 + (1.24 - 3.84i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-0.139 + 0.101i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.864 - 0.627i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.140 - 0.432i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 0.873T + 23T^{2} \)
29 \( 1 + (1.10 - 3.41i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (5.02 - 3.65i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.440 - 1.35i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.43 + 4.40i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 8.59T + 43T^{2} \)
47 \( 1 + (-3.75 - 11.5i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (2.31 - 1.67i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.0133 - 0.0409i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (4.92 + 3.58i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 11.3T + 67T^{2} \)
71 \( 1 + (9.34 + 6.78i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.581 + 1.79i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (6.76 - 4.91i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (8.92 + 6.48i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 11.5T + 89T^{2} \)
97 \( 1 + (5.38 - 3.91i)T + (29.9 - 92.2i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.145151206863054773422577803339, −8.884773143427413434258957564733, −8.120737439136670867006359168381, −7.16770052457733208134815687776, −6.13497774521148299105493022203, −5.64361622521481428755336375928, −4.75746326061576424962237129125, −3.48626824922824797745958317663, −2.88380137210337912653311538200, −1.53810983340365462821626052344, 0.18056460286297888478528172814, 1.63044149555205977453125139294, 3.01057962840940199893594230493, 3.94836565980103649845546337692, 4.50153525549255519174915848083, 5.65748394741780171995328450016, 6.79197350851331221884007745391, 7.18314995314831318130090735773, 7.83730837825291963600054266816, 8.860826411381365074044283546389

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