Properties

Degree $1$
Conductor $264$
Sign $1$
Motivic weight $0$
Primitive yes
Self-dual yes
Analytic rank $0$

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + 5-s + 7-s + 13-s + 17-s − 19-s + 23-s + 25-s − 29-s − 31-s + 35-s − 37-s + 41-s − 43-s + 47-s + 49-s + 53-s − 59-s + 61-s + 65-s + 67-s + 71-s − 73-s + 79-s + 83-s + 85-s − 89-s + 91-s + ⋯
L(s,χ)  = 1  + 5-s + 7-s + 13-s + 17-s − 19-s + 23-s + 25-s − 29-s − 31-s + 35-s − 37-s + 41-s − 43-s + 47-s + 49-s + 53-s − 59-s + 61-s + 65-s + 67-s + 71-s − 73-s + 79-s + 83-s + 85-s − 89-s + 91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 264 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(264\)    =    \(2^{3} \cdot 3 \cdot 11\)
Sign: $1$
Motivic weight: \(0\)
Character: $\chi_{264} (131, \cdot )$
Sato-Tate group: $\mu(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 264,\ (1:\ ),\ 1)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(2.844548380\)
\(L(\frac12,\chi)\) \(\approx\) \(2.844548380\)
\(L(\chi,1)\) \(\approx\) \(1.546813295\)
\(L(1,\chi)\) \(\approx\) \(1.546813295\)

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

−25.531960448537157631594276839459, −24.829157847397313069835967068313, −23.79704343202651028015311554008, −22.96427397529695889492662649037, −21.70549995786776940028664026649, −21.058147923555650251209338233908, −20.45387603207282371794004464792, −18.91959095759050338527810879680, −18.23150594034769273386957970013, −17.265927884604002724014016745818, −16.58327959601650045146752714511, −15.139373570533744946321155565089, −14.3742580244394296721147910294, −13.4640543843162528218905202283, −12.524746174620956357801102927243, −11.167279887507364811712729873020, −10.50622664166826051946269658673, −9.22851836057645568838227204128, −8.38920597741791808642770967390, −7.13580286638191258357763543918, −5.88686609056203021622173216366, −5.097855274008953324543337648500, −3.691154311760920389665436180617, −2.16453672480886159225963517091, −1.16648103238616476814008084455, 1.16648103238616476814008084455, 2.16453672480886159225963517091, 3.691154311760920389665436180617, 5.097855274008953324543337648500, 5.88686609056203021622173216366, 7.13580286638191258357763543918, 8.38920597741791808642770967390, 9.22851836057645568838227204128, 10.50622664166826051946269658673, 11.167279887507364811712729873020, 12.524746174620956357801102927243, 13.4640543843162528218905202283, 14.3742580244394296721147910294, 15.139373570533744946321155565089, 16.58327959601650045146752714511, 17.265927884604002724014016745818, 18.23150594034769273386957970013, 18.91959095759050338527810879680, 20.45387603207282371794004464792, 21.058147923555650251209338233908, 21.70549995786776940028664026649, 22.96427397529695889492662649037, 23.79704343202651028015311554008, 24.829157847397313069835967068313, 25.531960448537157631594276839459

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