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Properties

Label 2-1980-11.3-c1-0-5
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

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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

\(L(1)\) \(\approx\) \(1.531196692\)
\(L(\frac12)\) \(\approx\) \(1.531196692\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.93 - 3.58i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.276985322616214697453042239923, −8.461141328559564110124334185515, −7.69831152446576869080267820837, −7.04663464434670751659108006428, −6.04166663750946895584055111255, −5.32783317844695338158608852011, −4.02364901966164813750744685762, −3.83626349758318282854714076893, −2.24998932514533957645204054012, −1.15081625920631282185243098767, 0.63853773496039671208623388393, 2.15436860369261766578021466653, 3.19802959280764197380568771928, 4.08272335309575981692118636231, 4.98457843175693975343166005406, 5.97964017685449074321082817697, 6.69945468366357678913249301721, 7.53633162852674196088773392829, 8.234754776378062193036664958946, 9.196177864029237204307084573954

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