Properties

Degree 1
Conductor $ 2^{2} \cdot 29 $
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  + 3-s + 5-s − 7-s + 9-s + 11-s + 13-s + 15-s − 17-s + 19-s − 21-s − 23-s + 25-s + 27-s + 31-s + 33-s − 35-s − 37-s + 39-s − 41-s + 43-s + 45-s + 47-s + 49-s − 51-s + 53-s + 55-s + 57-s + ⋯
L(s,χ)  = 1  + 3-s + 5-s − 7-s + 9-s + 11-s + 13-s + 15-s − 17-s + 19-s − 21-s − 23-s + 25-s + 27-s + 31-s + 33-s − 35-s − 37-s + 39-s − 41-s + 43-s + 45-s + 47-s + 49-s − 51-s + 53-s + 55-s + 57-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 116 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(116\)    =    \(2^{2} \cdot 29\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{116} (115, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(1,\ 116,\ (1:\ ),\ 1)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.884518775$
$L(\frac12,\chi)$  $\approx$  $2.884518775$
$L(\chi,1)$  $\approx$  1.750137330
$L(1,\chi)$  $\approx$  1.750137330

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−29.10728884062669449984644030521, −28.05817943106239695976732655940, −26.59677823869733120585123080170, −25.91057950875139359531491722626, −25.094720800369474213100059780532, −24.29455239315401670929083628302, −22.58255821231777735983801196344, −21.839864950721796025022224864965, −20.63057804260674935785541900476, −19.83502358996123754453981580244, −18.747444327813632650386305438473, −17.7010932101502087834912320818, −16.31970268674881718170052687995, −15.35734320320145801398011847233, −13.82937879351773359600082980971, −13.59278038210108867253298124701, −12.211062797860126232968613635788, −10.411402146226654700457772204130, −9.415397688391975929238262604479, −8.701173952921560494904822111532, −6.96956393553234557071638797738, −6.017382235992170490422307344071, −4.08103659546516473538382287793, −2.868371820651405764455340610178, −1.452918860738990389442431257107, 1.452918860738990389442431257107, 2.868371820651405764455340610178, 4.08103659546516473538382287793, 6.017382235992170490422307344071, 6.96956393553234557071638797738, 8.701173952921560494904822111532, 9.415397688391975929238262604479, 10.411402146226654700457772204130, 12.211062797860126232968613635788, 13.59278038210108867253298124701, 13.82937879351773359600082980971, 15.35734320320145801398011847233, 16.31970268674881718170052687995, 17.7010932101502087834912320818, 18.747444327813632650386305438473, 19.83502358996123754453981580244, 20.63057804260674935785541900476, 21.839864950721796025022224864965, 22.58255821231777735983801196344, 24.29455239315401670929083628302, 25.094720800369474213100059780532, 25.91057950875139359531491722626, 26.59677823869733120585123080170, 28.05817943106239695976732655940, 29.10728884062669449984644030521

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