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Properties

Label 2-1980-11.3-c1-0-10
Degree $2$
Conductor $1980$
Sign $0.999 - 0.0335i$
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 + (0.900 − 2.77i)7-s + (1.20 + 3.09i)11-s + (1.10 − 0.800i)13-s + (4.44 + 3.23i)17-s + (−0.402 − 1.23i)19-s − 1.82·23-s + (0.309 + 0.951i)25-s + (−1.16 + 3.59i)29-s + (1.99 − 1.44i)31-s + (2.35 − 1.71i)35-s + (1.02 − 3.15i)37-s + (1.25 + 3.85i)41-s + 5.61·43-s + (−1.19 − 3.67i)47-s + ⋯
L(s)  = 1  + (0.361 + 0.262i)5-s + (0.340 − 1.04i)7-s + (0.362 + 0.931i)11-s + (0.305 − 0.222i)13-s + (1.07 + 0.783i)17-s + (−0.0923 − 0.284i)19-s − 0.381·23-s + (0.0618 + 0.190i)25-s + (−0.216 + 0.666i)29-s + (0.357 − 0.259i)31-s + (0.398 − 0.289i)35-s + (0.168 − 0.518i)37-s + (0.195 + 0.602i)41-s + 0.856·43-s + (−0.174 − 0.536i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0335i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0335i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1980\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $0.999 - 0.0335i$
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: \((2,\ 1980,\ (\ :1/2),\ 0.999 - 0.0335i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.105599829\)
\(L(\frac12)\) \(\approx\) \(2.105599829\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.20 - 3.09i)T \)
good7 \( 1 + (-0.900 + 2.77i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-1.10 + 0.800i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-4.44 - 3.23i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.402 + 1.23i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 1.82T + 23T^{2} \)
29 \( 1 + (1.16 - 3.59i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.99 + 1.44i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.02 + 3.15i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.25 - 3.85i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 5.61T + 43T^{2} \)
47 \( 1 + (1.19 + 3.67i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (3.12 - 2.26i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-3.44 + 10.5i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-1.18 - 0.858i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 11.7T + 67T^{2} \)
71 \( 1 + (-2.68 - 1.94i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-3.74 + 11.5i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (9.61 - 6.98i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-0.386 - 0.280i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 14.0T + 89T^{2} \)
97 \( 1 + (-3.11 + 2.26i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.349084797060711849331006710777, −8.232536942335053732335912581029, −7.60415007779282287776379907250, −6.87293020190511306349495311396, −6.07492657517028155813933230326, −5.11144457356047618869776432519, −4.17886064326297906542976287807, −3.45105743113091844035583278371, −2.09851774678717507718092694447, −1.06160961903689834358303122607, 1.00404273291095060314683518234, 2.23027969612467067069235751664, 3.19638300349737339488723605757, 4.26865529935392571762060899207, 5.41674643794132184894522623730, 5.78201663957097057373746751879, 6.66444059200196410335974378192, 7.81537497923212136807791334144, 8.434054436621854978521008959614, 9.142723523130637532351121321467

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