Properties

Label 2-3920-1.1-c1-0-20
Degree $2$
Conductor $3920$
Sign $1$
Analytic cond. $31.3013$
Root an. cond. $5.59476$
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  − 2.87·3-s + 5-s + 5.29·9-s − 1.46·11-s + 2.22·13-s − 2.87·15-s + 7.26·17-s + 5.48·19-s − 2.51·23-s + 25-s − 6.60·27-s − 7.12·29-s + 6.51·31-s + 4.22·33-s + 6.90·37-s − 6.39·39-s − 11.3·41-s − 3.31·43-s + 5.29·45-s + 8.36·47-s − 20.9·51-s − 7.00·53-s − 1.46·55-s − 15.8·57-s − 9.07·59-s + 11.2·61-s + 2.22·65-s + ⋯
L(s)  = 1  − 1.66·3-s + 0.447·5-s + 1.76·9-s − 0.441·11-s + 0.616·13-s − 0.743·15-s + 1.76·17-s + 1.25·19-s − 0.524·23-s + 0.200·25-s − 1.27·27-s − 1.32·29-s + 1.17·31-s + 0.734·33-s + 1.13·37-s − 1.02·39-s − 1.76·41-s − 0.505·43-s + 0.789·45-s + 1.22·47-s − 2.93·51-s − 0.962·53-s − 0.197·55-s − 2.09·57-s − 1.18·59-s + 1.43·61-s + 0.275·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(31.3013\)
Root analytic conductor: \(5.59476\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.210617433\)
\(L(\frac12)\) \(\approx\) \(1.210617433\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.87T + 3T^{2} \)
11 \( 1 + 1.46T + 11T^{2} \)
13 \( 1 - 2.22T + 13T^{2} \)
17 \( 1 - 7.26T + 17T^{2} \)
19 \( 1 - 5.48T + 19T^{2} \)
23 \( 1 + 2.51T + 23T^{2} \)
29 \( 1 + 7.12T + 29T^{2} \)
31 \( 1 - 6.51T + 31T^{2} \)
37 \( 1 - 6.90T + 37T^{2} \)
41 \( 1 + 11.3T + 41T^{2} \)
43 \( 1 + 3.31T + 43T^{2} \)
47 \( 1 - 8.36T + 47T^{2} \)
53 \( 1 + 7.00T + 53T^{2} \)
59 \( 1 + 9.07T + 59T^{2} \)
61 \( 1 - 11.2T + 61T^{2} \)
67 \( 1 - 6.41T + 67T^{2} \)
71 \( 1 - 10.5T + 71T^{2} \)
73 \( 1 - 10.5T + 73T^{2} \)
79 \( 1 + 10.1T + 79T^{2} \)
83 \( 1 + 16.4T + 83T^{2} \)
89 \( 1 - 9.83T + 89T^{2} \)
97 \( 1 + 2.09T + 97T^{2} \)
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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

−8.283319212062785850138319924322, −7.63009293203899413956179242413, −6.80991099645629661393873985052, −6.06605111410860563218184223885, −5.46621850197378854173999521811, −5.12703677309654570573831727374, −4.00057873257058881550515434115, −3.04884443364127360126484763651, −1.58335000728241232489185476295, −0.74151820273403776564704796328, 0.74151820273403776564704796328, 1.58335000728241232489185476295, 3.04884443364127360126484763651, 4.00057873257058881550515434115, 5.12703677309654570573831727374, 5.46621850197378854173999521811, 6.06605111410860563218184223885, 6.80991099645629661393873985052, 7.63009293203899413956179242413, 8.283319212062785850138319924322

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