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Properties

Label 1-683-683.682-r1-0-0
Degree $1$
Conductor $683$
Sign $1$
Analytic cond. $73.3985$
Root an. cond. $73.3985$
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 + 3-s + 4-s − 5-s − 6-s − 7-s − 8-s + 9-s + 10-s − 11-s + 12-s − 13-s + 14-s − 15-s + 16-s − 17-s − 18-s + 19-s − 20-s − 21-s + 22-s − 23-s − 24-s + 25-s + 26-s + 27-s − 28-s + ⋯
L(s)  = 1  − 2-s + 3-s + 4-s − 5-s − 6-s − 7-s − 8-s + 9-s + 10-s − 11-s + 12-s − 13-s + 14-s − 15-s + 16-s − 17-s − 18-s + 19-s − 20-s − 21-s + 22-s − 23-s − 24-s + 25-s + 26-s + 27-s − 28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 683 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 683 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(683\)
Sign: $1$
Analytic conductor: \(73.3985\)
Root analytic conductor: \(73.3985\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{683} (682, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 683,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6809904151\)
\(L(\frac12)\) \(\approx\) \(0.6809904151\)
\(L(1)\) \(\approx\) \(0.6010485104\)
\(L(1)\) \(\approx\) \(0.6010485104\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad683 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + T \)
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

−22.385799798697436728945559257036, −21.535132082608553416993451056515, −20.289913427438072961044843436771, −19.99865315813883315677173789365, −19.359713042782919603352869962552, −18.59080105750568856592709645465, −17.901936440994326364260890783700, −16.52462914478135192018629055249, −15.81368575333946580535740442039, −15.48438097114050326055749839774, −14.47873011017494453148424046920, −13.25745696276381454570625826462, −12.42731150249920021444232088941, −11.60401002503228439516069001862, −10.2966712103513931796100169157, −9.87665972823642925927117928707, −8.80372855203777052453598352784, −8.147866125211546943594823119353, −7.26640462038741175838117559075, −6.76258795480525762882405648371, −5.11300551475594820563681674840, −3.703943976193201134823572223127, −2.94428930360938590744425132706, −2.079252005885697762195421134746, −0.43040338128011344070994182028, 0.43040338128011344070994182028, 2.079252005885697762195421134746, 2.94428930360938590744425132706, 3.703943976193201134823572223127, 5.11300551475594820563681674840, 6.76258795480525762882405648371, 7.26640462038741175838117559075, 8.147866125211546943594823119353, 8.80372855203777052453598352784, 9.87665972823642925927117928707, 10.2966712103513931796100169157, 11.60401002503228439516069001862, 12.42731150249920021444232088941, 13.25745696276381454570625826462, 14.47873011017494453148424046920, 15.48438097114050326055749839774, 15.81368575333946580535740442039, 16.52462914478135192018629055249, 17.901936440994326364260890783700, 18.59080105750568856592709645465, 19.359713042782919603352869962552, 19.99865315813883315677173789365, 20.289913427438072961044843436771, 21.535132082608553416993451056515, 22.385799798697436728945559257036

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