Properties

Label 2-1960-1.1-c1-0-3
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  − 3-s + 5-s − 2·9-s − 5·11-s − 7·13-s − 15-s + 3·17-s + 2·19-s + 8·23-s + 25-s + 5·27-s − 5·29-s + 10·31-s + 5·33-s + 4·37-s + 7·39-s + 6·41-s + 2·43-s − 2·45-s + 7·47-s − 3·51-s − 10·53-s − 5·55-s − 2·57-s + 10·59-s + 12·61-s − 7·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s − 2/3·9-s − 1.50·11-s − 1.94·13-s − 0.258·15-s + 0.727·17-s + 0.458·19-s + 1.66·23-s + 1/5·25-s + 0.962·27-s − 0.928·29-s + 1.79·31-s + 0.870·33-s + 0.657·37-s + 1.12·39-s + 0.937·41-s + 0.304·43-s − 0.298·45-s + 1.02·47-s − 0.420·51-s − 1.37·53-s − 0.674·55-s − 0.264·57-s + 1.30·59-s + 1.53·61-s − 0.868·65-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

\(L(1)\) \(\approx\) \(1.087245619\)
\(L(\frac12)\) \(\approx\) \(1.087245619\)
\(L(\frac{3}{2})\) 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 \( 1 + 7 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 7 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
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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

−9.402195175064196345224163923961, −8.288225635235158209049417531779, −7.56793846342897653210506471906, −6.88382824698342573621557941164, −5.70455601651800109248396826390, −5.28603537534421022550155880259, −4.62651714719306486614775648654, −2.91065351671621099691740507495, −2.51581326623504893362388190719, −0.69036005295889719279144291289, 0.69036005295889719279144291289, 2.51581326623504893362388190719, 2.91065351671621099691740507495, 4.62651714719306486614775648654, 5.28603537534421022550155880259, 5.70455601651800109248396826390, 6.88382824698342573621557941164, 7.56793846342897653210506471906, 8.288225635235158209049417531779, 9.402195175064196345224163923961

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