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Properties

Label 1-2643-2643.2642-r1-0-0
Degree $1$
Conductor $2643$
Sign $1$
Analytic cond. $284.029$
Root an. cond. $284.029$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s − 11-s + 13-s + 14-s + 16-s + 17-s + 19-s − 20-s + 22-s + 23-s + 25-s − 26-s − 28-s + 29-s − 31-s − 32-s − 34-s + 35-s + 37-s − 38-s + 40-s − 41-s + ⋯
L(s)  = 1  − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s − 11-s + 13-s + 14-s + 16-s + 17-s + 19-s − 20-s + 22-s + 23-s + 25-s − 26-s − 28-s + 29-s − 31-s − 32-s − 34-s + 35-s + 37-s − 38-s + 40-s − 41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2643 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2643 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2643\)    =    \(3 \cdot 881\)
Sign: $1$
Analytic conductor: \(284.029\)
Root analytic conductor: \(284.029\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2643} (2642, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 2643,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.077450010\)
\(L(\frac12)\) \(\approx\) \(1.077450010\)
\(L(1)\) \(\approx\) \(0.6110845324\)
\(L(1)\) \(\approx\) \(0.6110845324\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
881 \( 1 \)
good2 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 - T \)
13 \( 1 + T \)
17 \( 1 + T \)
19 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 + T \)
31 \( 1 - T \)
37 \( 1 + T \)
41 \( 1 - T \)
43 \( 1 + T \)
47 \( 1 + T \)
53 \( 1 + T \)
59 \( 1 + T \)
61 \( 1 + T \)
67 \( 1 + T \)
71 \( 1 - T \)
73 \( 1 - T \)
79 \( 1 - T \)
83 \( 1 - T \)
89 \( 1 - T \)
97 \( 1 + T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.897914871497949322316520692739, −18.68753385276458544428573459301, −17.96641724407880939949357813629, −16.89195398144477851640387210399, −16.20507192323334543855295393096, −15.89708363164223644159420702687, −15.28654102931575465954051151967, −14.376733091663318730473680573492, −13.21675616782886125357784255095, −12.62227308340866361494047305242, −11.8213730749324000352955795009, −11.16013488368680274813514701325, −10.404544522759412199424138813138, −9.822583074186930466294194220873, −8.85679688906677020970331428027, −8.34488374117498926756437264458, −7.38250460837006212443473318408, −7.10344812607489451092570730139, −5.977848142759518797956716374728, −5.31502169452032608632333560171, −3.91997153541025748560276604288, −3.17510251151455063989860075020, −2.651316657087844101499237888230, −1.11748484947301266490337120181, −0.56153806064325738457272468488, 0.56153806064325738457272468488, 1.11748484947301266490337120181, 2.651316657087844101499237888230, 3.17510251151455063989860075020, 3.91997153541025748560276604288, 5.31502169452032608632333560171, 5.977848142759518797956716374728, 7.10344812607489451092570730139, 7.38250460837006212443473318408, 8.34488374117498926756437264458, 8.85679688906677020970331428027, 9.822583074186930466294194220873, 10.404544522759412199424138813138, 11.16013488368680274813514701325, 11.8213730749324000352955795009, 12.62227308340866361494047305242, 13.21675616782886125357784255095, 14.376733091663318730473680573492, 15.28654102931575465954051151967, 15.89708363164223644159420702687, 16.20507192323334543855295393096, 16.89195398144477851640387210399, 17.96641724407880939949357813629, 18.68753385276458544428573459301, 18.897914871497949322316520692739

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