Properties

Label 2-1960-1.1-c1-0-1
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.705·3-s − 5-s − 2.50·9-s − 4.50·11-s − 5.09·13-s + 0.705·15-s − 2·17-s + 3.09·19-s + 5.79·23-s + 25-s + 3.88·27-s + 9.50·29-s − 5.41·31-s + 3.17·33-s + 7.09·37-s + 3.59·39-s + 6.59·41-s + 4.70·43-s + 2.50·45-s − 10.0·47-s + 1.41·51-s + 9.91·53-s + 4.50·55-s − 2.18·57-s − 8·59-s + 8.91·61-s + 5.09·65-s + ⋯
L(s)  = 1  − 0.407·3-s − 0.447·5-s − 0.833·9-s − 1.35·11-s − 1.41·13-s + 0.182·15-s − 0.485·17-s + 0.709·19-s + 1.20·23-s + 0.200·25-s + 0.747·27-s + 1.76·29-s − 0.971·31-s + 0.553·33-s + 1.16·37-s + 0.575·39-s + 1.02·41-s + 0.717·43-s + 0.372·45-s − 1.47·47-s + 0.197·51-s + 1.36·53-s + 0.607·55-s − 0.288·57-s − 1.04·59-s + 1.14·61-s + 0.631·65-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1960,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8328627567\)
\(L(\frac12)\) \(\approx\) \(0.8328627567\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.705T + 3T^{2} \)
11 \( 1 + 4.50T + 11T^{2} \)
13 \( 1 + 5.09T + 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 3.09T + 19T^{2} \)
23 \( 1 - 5.79T + 23T^{2} \)
29 \( 1 - 9.50T + 29T^{2} \)
31 \( 1 + 5.41T + 31T^{2} \)
37 \( 1 - 7.09T + 37T^{2} \)
41 \( 1 - 6.59T + 41T^{2} \)
43 \( 1 - 4.70T + 43T^{2} \)
47 \( 1 + 10.0T + 47T^{2} \)
53 \( 1 - 9.91T + 53T^{2} \)
59 \( 1 + 8T + 59T^{2} \)
61 \( 1 - 8.91T + 61T^{2} \)
67 \( 1 + 0.117T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 8.18T + 73T^{2} \)
79 \( 1 + 14.1T + 79T^{2} \)
83 \( 1 - 10.7T + 83T^{2} \)
89 \( 1 + 4.09T + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−9.160468311340555659911197444602, −8.323274962431886353361338969336, −7.58371580045920996521435895857, −6.95834137157086356284612635905, −5.85817217685521681936364632804, −5.09466655157701723377513146181, −4.54831879709198772925321857535, −3.03137613377258161213430177920, −2.52060358696877775323471157822, −0.59160262913967686886142696613, 0.59160262913967686886142696613, 2.52060358696877775323471157822, 3.03137613377258161213430177920, 4.54831879709198772925321857535, 5.09466655157701723377513146181, 5.85817217685521681936364632804, 6.95834137157086356284612635905, 7.58371580045920996521435895857, 8.323274962431886353361338969336, 9.160468311340555659911197444602

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