Properties

Label 1-123-123.122-r1-0-0
Degree $1$
Conductor $123$
Sign $1$
Analytic cond. $13.2181$
Root an. cond. $13.2181$
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 + 43-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 + 43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(123\)    =    \(3 \cdot 41\)
Sign: $1$
Analytic conductor: \(13.2181\)
Root analytic conductor: \(13.2181\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{123} (122, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 123,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6945011863\)
\(L(\frac12)\) \(\approx\) \(0.6945011863\)
\(L(1)\) \(\approx\) \(0.5665357400\)
\(L(1)\) \(\approx\) \(0.5665357400\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
41 \( 1 \)
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 \( 1 + T \)
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

−28.496527443601652408427865635740, −27.59917777227529689878228166703, −26.89432462694290690808266950409, −25.83509616929602627444056089226, −24.938416984263702470065988871, −23.826188122962838716435547624147, −22.727924817343928308288707222516, −21.5204251909043245028683405365, −19.997861758964299517454572338119, −19.49391290582832650661058388718, −18.75126776465894076802508446038, −17.23025221714950748742520443585, −16.46317566449174461431645527985, −15.50350436889921861898463172209, −14.4471050579784769911305083960, −12.40809531095054900728934171, −11.888783154224987558623500060129, −10.43212103150766259393218738374, −9.479978199540395501709374256384, −8.28458807072094486078474133589, −7.19396503957577878369835475155, −6.178193289599403082810949075623, −4.04990216085087181767171763133, −2.70064330904441363956042357960, −0.69365918792444668323623063934, 0.69365918792444668323623063934, 2.70064330904441363956042357960, 4.04990216085087181767171763133, 6.178193289599403082810949075623, 7.19396503957577878369835475155, 8.28458807072094486078474133589, 9.479978199540395501709374256384, 10.43212103150766259393218738374, 11.888783154224987558623500060129, 12.40809531095054900728934171, 14.4471050579784769911305083960, 15.50350436889921861898463172209, 16.46317566449174461431645527985, 17.23025221714950748742520443585, 18.75126776465894076802508446038, 19.49391290582832650661058388718, 19.997861758964299517454572338119, 21.5204251909043245028683405365, 22.727924817343928308288707222516, 23.826188122962838716435547624147, 24.938416984263702470065988871, 25.83509616929602627444056089226, 26.89432462694290690808266950409, 27.59917777227529689878228166703, 28.496527443601652408427865635740

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