Properties

Label 2-1960-1.1-c1-0-23
Degree $2$
Conductor $1960$
Sign $-1$
Analytic cond. $15.6506$
Root an. cond. $3.95609$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 5-s + 9-s + 4·11-s − 2·13-s + 2·15-s − 2·19-s − 4·23-s + 25-s + 4·27-s + 10·29-s − 4·31-s − 8·33-s − 2·37-s + 4·39-s + 12·41-s − 4·43-s − 45-s − 4·47-s + 2·53-s − 4·55-s + 4·57-s − 10·59-s − 6·61-s + 2·65-s + 4·67-s + 8·69-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.447·5-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 0.516·15-s − 0.458·19-s − 0.834·23-s + 1/5·25-s + 0.769·27-s + 1.85·29-s − 0.718·31-s − 1.39·33-s − 0.328·37-s + 0.640·39-s + 1.87·41-s − 0.609·43-s − 0.149·45-s − 0.583·47-s + 0.274·53-s − 0.539·55-s + 0.529·57-s − 1.30·59-s − 0.768·61-s + 0.248·65-s + 0.488·67-s + 0.963·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1960\)    =    \(2^{3} \cdot 5 \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(15.6506\)
Root analytic conductor: \(3.95609\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1960} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1960,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−8.797314786123979806993947256863, −7.984654042637741729210238361128, −6.98510787646150277319641738743, −6.39409529064572332277892048898, −5.68272740720851408885613690946, −4.67179696959216786374064647679, −4.07506500324096665697518707967, −2.81457574838920705903756418297, −1.32595961593316370703892657095, 0, 1.32595961593316370703892657095, 2.81457574838920705903756418297, 4.07506500324096665697518707967, 4.67179696959216786374064647679, 5.68272740720851408885613690946, 6.39409529064572332277892048898, 6.98510787646150277319641738743, 7.984654042637741729210238361128, 8.797314786123979806993947256863

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