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Properties

Label 2-2007-223.52-c0-0-0
Degree $2$
Conductor $2007$
Sign $-0.221 + 0.975i$
Analytic cond. $1.00162$
Root an. cond. $1.00081$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.873 − 0.487i)4-s + (0.307 − 1.00i)7-s + (0.869 − 1.07i)13-s + (0.524 + 0.851i)16-s + (−1.28 + 0.871i)19-s + (0.0424 − 0.999i)25-s + (−0.757 + 0.725i)28-s + (−0.492 − 0.559i)31-s + (−1.20 − 0.544i)37-s + (−0.0703 − 1.65i)43-s + (−0.0825 − 0.0557i)49-s + (−1.28 + 0.515i)52-s + (1.20 − 1.48i)61-s + (−0.0424 − 0.999i)64-s + (−0.164 + 0.295i)67-s + ⋯
L(s)  = 1  + (−0.873 − 0.487i)4-s + (0.307 − 1.00i)7-s + (0.869 − 1.07i)13-s + (0.524 + 0.851i)16-s + (−1.28 + 0.871i)19-s + (0.0424 − 0.999i)25-s + (−0.757 + 0.725i)28-s + (−0.492 − 0.559i)31-s + (−1.20 − 0.544i)37-s + (−0.0703 − 1.65i)43-s + (−0.0825 − 0.0557i)49-s + (−1.28 + 0.515i)52-s + (1.20 − 1.48i)61-s + (−0.0424 − 0.999i)64-s + (−0.164 + 0.295i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2007\)    =    \(3^{2} \cdot 223\)
Sign: $-0.221 + 0.975i$
Analytic conductor: \(1.00162\)
Root analytic conductor: \(1.00081\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2007} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2007,\ (\ :0),\ -0.221 + 0.975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8395983206\)
\(L(\frac12)\) \(\approx\) \(0.8395983206\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
223 \( 1 + (0.0424 - 0.999i)T \)
good2 \( 1 + (0.873 + 0.487i)T^{2} \)
5 \( 1 + (-0.0424 + 0.999i)T^{2} \)
7 \( 1 + (-0.307 + 1.00i)T + (-0.828 - 0.559i)T^{2} \)
11 \( 1 + (-0.660 + 0.750i)T^{2} \)
13 \( 1 + (-0.869 + 1.07i)T + (-0.210 - 0.977i)T^{2} \)
17 \( 1 + (0.660 - 0.750i)T^{2} \)
19 \( 1 + (1.28 - 0.871i)T + (0.372 - 0.927i)T^{2} \)
23 \( 1 + (-0.0424 + 0.999i)T^{2} \)
29 \( 1 + (-0.450 + 0.892i)T^{2} \)
31 \( 1 + (0.492 + 0.559i)T + (-0.127 + 0.991i)T^{2} \)
37 \( 1 + (1.20 + 0.544i)T + (0.660 + 0.750i)T^{2} \)
41 \( 1 + (0.778 - 0.628i)T^{2} \)
43 \( 1 + (0.0703 + 1.65i)T + (-0.996 + 0.0848i)T^{2} \)
47 \( 1 + (0.372 + 0.927i)T^{2} \)
53 \( 1 + (-0.292 - 0.956i)T^{2} \)
59 \( 1 + (-0.942 - 0.333i)T^{2} \)
61 \( 1 + (-1.20 + 1.48i)T + (-0.210 - 0.977i)T^{2} \)
67 \( 1 + (0.164 - 0.295i)T + (-0.524 - 0.851i)T^{2} \)
71 \( 1 + (-0.942 - 0.333i)T^{2} \)
73 \( 1 + (0.651 + 1.62i)T + (-0.721 + 0.691i)T^{2} \)
79 \( 1 + (-0.665 - 1.88i)T + (-0.778 + 0.628i)T^{2} \)
83 \( 1 + (-0.996 - 0.0848i)T^{2} \)
89 \( 1 + (0.0424 + 0.999i)T^{2} \)
97 \( 1 + (-0.461 + 0.481i)T + (-0.0424 - 0.999i)T^{2} \)
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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

−9.015731605625150865506624633543, −8.352428308790564280530847342570, −7.78991099369892308368319751762, −6.67195736205034761784570697065, −5.85973236683445193823181307728, −5.10141517748386256769615190262, −4.06849240618426092189462471477, −3.64221938256447821841536554687, −1.93485120448786111484576762745, −0.64515396232922760850636254463, 1.65590700861249534083242191341, 2.86551489478915879369324101056, 3.89281426322994197489868372777, 4.68356611649867921427490533232, 5.45461871705482107244330014152, 6.41422585875534175573198322043, 7.24817598584451436489439091392, 8.340688005748156172516201382636, 8.830801669675467532798343413557, 9.178683550957799786768229409920

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