Properties

Label 2-1980-1.1-c1-0-16
Degree $2$
Conductor $1980$
Sign $-1$
Analytic cond. $15.8103$
Root an. cond. $3.97622$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 4.60·7-s − 11-s − 4.60·13-s − 6.60·17-s − 7.21·19-s + 25-s − 8·29-s + 9.21·31-s − 4.60·35-s − 3.21·37-s − 8·41-s − 3.39·43-s + 5.21·47-s + 14.2·49-s − 2·53-s + 55-s − 8·59-s + 7.21·61-s + 4.60·65-s − 4·67-s + 14.4·71-s + 0.605·73-s − 4.60·77-s − 11.2·79-s + 10.6·83-s + 6.60·85-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.74·7-s − 0.301·11-s − 1.27·13-s − 1.60·17-s − 1.65·19-s + 0.200·25-s − 1.48·29-s + 1.65·31-s − 0.778·35-s − 0.527·37-s − 1.24·41-s − 0.517·43-s + 0.760·47-s + 2.03·49-s − 0.274·53-s + 0.134·55-s − 1.04·59-s + 0.923·61-s + 0.571·65-s − 0.488·67-s + 1.71·71-s + 0.0708·73-s − 0.524·77-s − 1.26·79-s + 1.16·83-s + 0.716·85-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
11 \( 1 + T \)
good7 \( 1 - 4.60T + 7T^{2} \)
13 \( 1 + 4.60T + 13T^{2} \)
17 \( 1 + 6.60T + 17T^{2} \)
19 \( 1 + 7.21T + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 8T + 29T^{2} \)
31 \( 1 - 9.21T + 31T^{2} \)
37 \( 1 + 3.21T + 37T^{2} \)
41 \( 1 + 8T + 41T^{2} \)
43 \( 1 + 3.39T + 43T^{2} \)
47 \( 1 - 5.21T + 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 + 8T + 59T^{2} \)
61 \( 1 - 7.21T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 - 14.4T + 71T^{2} \)
73 \( 1 - 0.605T + 73T^{2} \)
79 \( 1 + 11.2T + 79T^{2} \)
83 \( 1 - 10.6T + 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + 12.4T + 97T^{2} \)
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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

−8.472548828612932207190264938309, −8.197335054999536746745911895671, −7.27441162298511920027975522309, −6.57587484221464496548152404924, −5.30258562588128099124622267969, −4.65499860166824917316382249300, −4.10828116922742993371669978566, −2.50578319035411940419283198700, −1.81473950808663865241627091514, 0, 1.81473950808663865241627091514, 2.50578319035411940419283198700, 4.10828116922742993371669978566, 4.65499860166824917316382249300, 5.30258562588128099124622267969, 6.57587484221464496548152404924, 7.27441162298511920027975522309, 8.197335054999536746745911895671, 8.472548828612932207190264938309

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