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Properties

Label 2-1980-11.3-c1-0-16
Degree $2$
Conductor $1980$
Sign $-0.0180 + 0.999i$
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 + (0.714 − 2.19i)7-s + (3.14 − 1.04i)11-s + (4.18 − 3.04i)13-s + (0.339 + 0.246i)17-s + (−0.537 − 1.65i)19-s − 4.90·23-s + (0.309 + 0.951i)25-s + (0.849 − 2.61i)29-s + (−5.30 + 3.85i)31-s + (−1.87 + 1.35i)35-s + (−3.22 + 9.93i)37-s + (−2.61 − 8.03i)41-s + 6.16·43-s + (0.320 + 0.987i)47-s + ⋯
L(s)  = 1  + (−0.361 − 0.262i)5-s + (0.269 − 0.830i)7-s + (0.949 − 0.314i)11-s + (1.16 − 0.843i)13-s + (0.0824 + 0.0598i)17-s + (−0.123 − 0.379i)19-s − 1.02·23-s + (0.0618 + 0.190i)25-s + (0.157 − 0.485i)29-s + (−0.953 + 0.692i)31-s + (−0.316 + 0.229i)35-s + (−0.530 + 1.63i)37-s + (−0.407 − 1.25i)41-s + 0.940·43-s + (0.0467 + 0.144i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0180 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0180 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1980\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $-0.0180 + 0.999i$
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: \((2,\ 1980,\ (\ :1/2),\ -0.0180 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.635351289\)
\(L(\frac12)\) \(\approx\) \(1.635351289\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.14 + 1.04i)T \)
good7 \( 1 + (-0.714 + 2.19i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-4.18 + 3.04i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.339 - 0.246i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.537 + 1.65i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 4.90T + 23T^{2} \)
29 \( 1 + (-0.849 + 2.61i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (5.30 - 3.85i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (3.22 - 9.93i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.61 + 8.03i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 6.16T + 43T^{2} \)
47 \( 1 + (-0.320 - 0.987i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-4.38 + 3.18i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-4.45 + 13.7i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (10.2 + 7.47i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 0.559T + 67T^{2} \)
71 \( 1 + (-4.17 - 3.03i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-3.83 + 11.8i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-3.91 + 2.84i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (5.83 + 4.23i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 8.95T + 89T^{2} \)
97 \( 1 + (-0.633 + 0.460i)T + (29.9 - 92.2i)T^{2} \)
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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

−8.789910060003984919018345905123, −8.268348235736366542435009598708, −7.47947760169220807884689898175, −6.60514501055300966835379052617, −5.85249064450821609782343885061, −4.82482165077834482853694025920, −3.87953934681193928053256524033, −3.36884578199487088835745861413, −1.69855812826382019260083985469, −0.63622079504630462210586210036, 1.42941349957106726775409564145, 2.40770084459948149363213636197, 3.78791367550993460009848415599, 4.19353811312852697106253724376, 5.54889416966868374967402872678, 6.14982285340747782350088777740, 7.00216075068797934905031407033, 7.81905589005106515495016493606, 8.824788769531758346246031690444, 9.060396249911950218246453325873

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