Properties

Degree $1$
Conductor $2219$
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 − 8-s + 9-s − 10-s + 11-s + 12-s + 13-s + 15-s + 16-s + 17-s − 18-s + 19-s + 20-s − 22-s + 23-s − 24-s + 25-s − 26-s + 27-s − 29-s − 30-s − 31-s − 32-s + ⋯
L(s,χ)  = 1  − 2-s + 3-s + 4-s + 5-s − 6-s − 8-s + 9-s − 10-s + 11-s + 12-s + 13-s + 15-s + 16-s + 17-s − 18-s + 19-s + 20-s − 22-s + 23-s − 24-s + 25-s − 26-s + 27-s − 29-s − 30-s − 31-s − 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 2219 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 2219 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2219\)    =    \(7 \cdot 317\)
Sign: $1$
Motivic weight: \(0\)
Character: $\chi_{2219} (2218, \cdot )$
Sato-Tate group: $\mu(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 2219,\ (1:\ ),\ 1)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(4.240811471\)
\(L(\frac12,\chi)\) \(\approx\) \(4.240811471\)
\(L(\chi,1)\) \(\approx\) \(1.600598712\)
\(L(1,\chi)\) \(\approx\) \(1.600598712\)

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

−19.498687693734588679893586248480, −18.61454793566656098033374454633, −18.38595077896106736999718159406, −17.45655703387417440318035871315, −16.664653144521369810654688315881, −16.16628617180335792614241292302, −15.136541007033498411689669918331, −14.52673065726978867056964566612, −13.89673443239879049702466142771, −13.032543363606489641736571932505, −12.279547919169806724395549328085, −11.20131017016852588268474413188, −10.556157517066609034506824766715, −9.5427119666342741664892369187, −9.29238476874812086511471640030, −8.699965201573715626741391203971, −7.63457882147232823150581657742, −7.1380807942850836689558146405, −6.13689266182158702579758688818, −5.50571643124434323538568704310, −4.00172007553361077719462787462, −3.1912422308395478175167562351, −2.434618813611404005868177228271, −1.30400146794627423483069024097, −1.1406550413878926907090100700, 1.1406550413878926907090100700, 1.30400146794627423483069024097, 2.434618813611404005868177228271, 3.1912422308395478175167562351, 4.00172007553361077719462787462, 5.50571643124434323538568704310, 6.13689266182158702579758688818, 7.1380807942850836689558146405, 7.63457882147232823150581657742, 8.699965201573715626741391203971, 9.29238476874812086511471640030, 9.5427119666342741664892369187, 10.556157517066609034506824766715, 11.20131017016852588268474413188, 12.279547919169806724395549328085, 13.032543363606489641736571932505, 13.89673443239879049702466142771, 14.52673065726978867056964566612, 15.136541007033498411689669918331, 16.16628617180335792614241292302, 16.664653144521369810654688315881, 17.45655703387417440318035871315, 18.38595077896106736999718159406, 18.61454793566656098033374454633, 19.498687693734588679893586248480

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