Properties

Label 2-1960-1.1-c1-0-30
Degree $2$
Conductor $1960$
Sign $-1$
Analytic cond. $15.6506$
Root an. cond. $3.95609$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.414·3-s − 5-s − 2.82·9-s + 0.828·11-s − 2·13-s + 0.414·15-s + 7.65·17-s + 5.65·19-s − 5.58·23-s + 25-s + 2.41·27-s − 7.82·29-s − 0.828·31-s − 0.343·33-s + 5.65·37-s + 0.828·39-s − 5.82·41-s − 6.89·43-s + 2.82·45-s − 11.6·47-s − 3.17·51-s − 5.65·53-s − 0.828·55-s − 2.34·57-s + 4·59-s − 6.65·61-s + 2·65-s + ⋯
L(s)  = 1  − 0.239·3-s − 0.447·5-s − 0.942·9-s + 0.249·11-s − 0.554·13-s + 0.106·15-s + 1.85·17-s + 1.29·19-s − 1.16·23-s + 0.200·25-s + 0.464·27-s − 1.45·29-s − 0.148·31-s − 0.0597·33-s + 0.929·37-s + 0.132·39-s − 0.910·41-s − 1.05·43-s + 0.421·45-s − 1.70·47-s − 0.444·51-s − 0.777·53-s − 0.111·55-s − 0.310·57-s + 0.520·59-s − 0.852·61-s + 0.248·65-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: no
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 \( 1 + 0.414T + 3T^{2} \)
11 \( 1 - 0.828T + 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 - 7.65T + 17T^{2} \)
19 \( 1 - 5.65T + 19T^{2} \)
23 \( 1 + 5.58T + 23T^{2} \)
29 \( 1 + 7.82T + 29T^{2} \)
31 \( 1 + 0.828T + 31T^{2} \)
37 \( 1 - 5.65T + 37T^{2} \)
41 \( 1 + 5.82T + 41T^{2} \)
43 \( 1 + 6.89T + 43T^{2} \)
47 \( 1 + 11.6T + 47T^{2} \)
53 \( 1 + 5.65T + 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 6.65T + 61T^{2} \)
67 \( 1 + 12.8T + 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + 3.65T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 4.75T + 83T^{2} \)
89 \( 1 - 5.34T + 89T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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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

−8.756695762399085299275253108086, −7.79314431319499445826951175320, −7.52086004847455421744693340558, −6.26712608188395108020021832776, −5.57402492917759257637210691215, −4.86710536399246228016371647026, −3.59707277010430743651647565967, −3.01806601408796427452784010423, −1.51676762360876229218550540439, 0, 1.51676762360876229218550540439, 3.01806601408796427452784010423, 3.59707277010430743651647565967, 4.86710536399246228016371647026, 5.57402492917759257637210691215, 6.26712608188395108020021832776, 7.52086004847455421744693340558, 7.79314431319499445826951175320, 8.756695762399085299275253108086

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