Properties

Label 2-1960-1.1-c1-0-37
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  + 3-s + 5-s − 2·9-s − 5·11-s − 13-s + 15-s − 3·17-s + 6·19-s − 6·23-s + 25-s − 5·27-s − 9·29-s − 5·33-s + 6·37-s − 39-s − 8·41-s + 6·43-s − 2·45-s − 3·47-s − 3·51-s − 12·53-s − 5·55-s + 6·57-s − 8·59-s + 4·61-s − 65-s − 4·67-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 2/3·9-s − 1.50·11-s − 0.277·13-s + 0.258·15-s − 0.727·17-s + 1.37·19-s − 1.25·23-s + 1/5·25-s − 0.962·27-s − 1.67·29-s − 0.870·33-s + 0.986·37-s − 0.160·39-s − 1.24·41-s + 0.914·43-s − 0.298·45-s − 0.437·47-s − 0.420·51-s − 1.64·53-s − 0.674·55-s + 0.794·57-s − 1.04·59-s + 0.512·61-s − 0.124·65-s − 0.488·67-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: Trivial
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 \( 1 - T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−8.860484777002540005828982336087, −7.74465662186675135601071019089, −7.69912855129361241271554591687, −6.28715004025144947864257791694, −5.55365867270428984617540263559, −4.85287109801086584074848656984, −3.56542759922310316766610018532, −2.71647306061986597321844269058, −1.93520907898928704565581532153, 0, 1.93520907898928704565581532153, 2.71647306061986597321844269058, 3.56542759922310316766610018532, 4.85287109801086584074848656984, 5.55365867270428984617540263559, 6.28715004025144947864257791694, 7.69912855129361241271554591687, 7.74465662186675135601071019089, 8.860484777002540005828982336087

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