Properties

Degree $1$
Conductor $421$
Sign $1$
Motivic weight $0$
Primitive yes
Self-dual yes
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 421 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 421 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(421\)
Sign: $1$
Motivic weight: \(0\)
Character: $\chi_{421} (420, \cdot )$
Sato-Tate group: $\mu(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 421,\ (0:\ ),\ 1)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(1.653457695\)
\(L(\frac12,\chi)\) \(\approx\) \(1.653457695\)
\(L(\chi,1)\) \(\approx\) \(1.267717333\)
\(L(1,\chi)\) \(\approx\) \(1.267717333\)

Euler product

   \(L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)
   \(L(s,\chi) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.67548666017703027948332037392, −23.873436895854448917638236567780, −22.045557771173145010342083924151, −21.31452708039016301137069253560, −20.67775897283310104023660054281, −19.84413318163877968858686962998, −18.99357098870415756784768188965, −18.20242984732001069195924852954, −17.22628867582552397451001718394, −16.77309856597790939993501125179, −15.280967490820672365391581943559, −14.55913732026156582272106023161, −14.00645629999339589445118944147, −12.58947355080643853416184958855, −11.68683867035796318843116813206, −10.32719512774026009780570311576, −9.79803260296559994417120836842, −8.871135029705840497296860001127, −8.11916428888506283005148788847, −7.17442730637807223611901308579, −6.14906965754500007346129628393, −4.76205400823331853705450145771, −3.27166034738286414706659734057, −1.995460235461726831162458753079, −1.55769232926572293821606083031, 1.55769232926572293821606083031, 1.995460235461726831162458753079, 3.27166034738286414706659734057, 4.76205400823331853705450145771, 6.14906965754500007346129628393, 7.17442730637807223611901308579, 8.11916428888506283005148788847, 8.871135029705840497296860001127, 9.79803260296559994417120836842, 10.32719512774026009780570311576, 11.68683867035796318843116813206, 12.58947355080643853416184958855, 14.00645629999339589445118944147, 14.55913732026156582272106023161, 15.280967490820672365391581943559, 16.77309856597790939993501125179, 17.22628867582552397451001718394, 18.20242984732001069195924852954, 18.99357098870415756784768188965, 19.84413318163877968858686962998, 20.67775897283310104023660054281, 21.31452708039016301137069253560, 22.045557771173145010342083924151, 23.873436895854448917638236567780, 24.67548666017703027948332037392

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