Properties

Label 2-1960-1.1-c1-0-8
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 + 2·11-s − 15-s − 4·17-s + 2·19-s + 23-s + 25-s + 5·27-s + 9·29-s − 4·31-s − 2·33-s + 4·37-s − 41-s + 9·43-s − 2·45-s + 4·51-s − 10·53-s + 2·55-s − 2·57-s + 10·59-s − 9·61-s + 5·67-s − 69-s + 14·71-s − 12·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s − 2/3·9-s + 0.603·11-s − 0.258·15-s − 0.970·17-s + 0.458·19-s + 0.208·23-s + 1/5·25-s + 0.962·27-s + 1.67·29-s − 0.718·31-s − 0.348·33-s + 0.657·37-s − 0.156·41-s + 1.37·43-s − 0.298·45-s + 0.560·51-s − 1.37·53-s + 0.269·55-s − 0.264·57-s + 1.30·59-s − 1.15·61-s + 0.610·67-s − 0.120·69-s + 1.66·71-s − 1.40·73-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

\(L(1)\) \(\approx\) \(1.426423692\)
\(L(\frac12)\) \(\approx\) \(1.426423692\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + T + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 9 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 11 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−9.126447015637667693088500493602, −8.577853488051574153517458260879, −7.55197449397548355810730048611, −6.54726694118631653414664331561, −6.13381189947026461337417746669, −5.18289036336620639563064974665, −4.44525519825866077450144641008, −3.23526529985027086432912393956, −2.22050794547907166274142371283, −0.827091807844068581217059306961, 0.827091807844068581217059306961, 2.22050794547907166274142371283, 3.23526529985027086432912393956, 4.44525519825866077450144641008, 5.18289036336620639563064974665, 6.13381189947026461337417746669, 6.54726694118631653414664331561, 7.55197449397548355810730048611, 8.577853488051574153517458260879, 9.126447015637667693088500493602

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