Properties

Label 2-1960-1.1-c1-0-0
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 5-s + 9-s − 11-s − 3·13-s + 2·15-s − 2·17-s − 5·19-s + 7·23-s + 25-s + 4·27-s − 6·29-s + 4·31-s + 2·33-s − 5·37-s + 6·39-s − 5·41-s + 6·43-s − 45-s − 9·47-s + 4·51-s + 11·53-s + 55-s + 10·57-s + 8·59-s − 12·61-s + 3·65-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.447·5-s + 1/3·9-s − 0.301·11-s − 0.832·13-s + 0.516·15-s − 0.485·17-s − 1.14·19-s + 1.45·23-s + 1/5·25-s + 0.769·27-s − 1.11·29-s + 0.718·31-s + 0.348·33-s − 0.821·37-s + 0.960·39-s − 0.780·41-s + 0.914·43-s − 0.149·45-s − 1.31·47-s + 0.560·51-s + 1.51·53-s + 0.134·55-s + 1.32·57-s + 1.04·59-s − 1.53·61-s + 0.372·65-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: \(0\)
Selberg data: \((2,\ 1960,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6227376888\)
\(L(\frac12)\) \(\approx\) \(0.6227376888\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 - 11 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−9.120494019087879646941493450871, −8.447359674641968247307299532517, −7.41175041159849215065524714179, −6.78703353203608464553115045216, −5.99797434225824589956437343584, −5.05403415737104360436911227846, −4.59475108981645282369545843095, −3.36085533604321633795096739870, −2.17010984041333828186019700181, −0.53684007895046932778022767435, 0.53684007895046932778022767435, 2.17010984041333828186019700181, 3.36085533604321633795096739870, 4.59475108981645282369545843095, 5.05403415737104360436911227846, 5.99797434225824589956437343584, 6.78703353203608464553115045216, 7.41175041159849215065524714179, 8.447359674641968247307299532517, 9.120494019087879646941493450871

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