Properties

Degree 1
Conductor $ 7 \cdot 13 $
Sign $-0.711 + 0.702i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s − 5-s + (0.5 + 0.866i)6-s + 8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s − 12-s + (−0.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + 18-s + (0.5 + 0.866i)19-s + (0.5 + 0.866i)20-s + ⋯
L(s,χ)  = 1  + (−0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s − 5-s + (0.5 + 0.866i)6-s + 8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s − 12-s + (−0.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + 18-s + (0.5 + 0.866i)19-s + (0.5 + 0.866i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.711 + 0.702i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.711 + 0.702i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(91\)    =    \(7 \cdot 13\)
\( \varepsilon \)  =  $-0.711 + 0.702i$
motivic weight  =  \(0\)
character  :  $\chi_{91} (55, \cdot )$
Sato-Tate  :  $\mu(6)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 91,\ (1:\ ),\ -0.711 + 0.702i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.2050940551 + 0.4996656788i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.2050940551 + 0.4996656788i\)
\(L(\chi,1)\)  \(\approx\)  \(0.6351074874 + 0.1745486873i\)
\(L(1,\chi)\)  \(\approx\)  \(0.6351074874 + 0.1745486873i\)

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

−29.78124650129052384624522523638, −28.40175559689678680464343590858, −27.680809327164464268674829444559, −26.65370353171126419586440247894, −26.24986034265509903875166830233, −24.64932636560842202546404322955, −23.05509519575281694541544477966, −22.07311875991833254721629121566, −21.01320372326687910170974199535, −20.14013726870933120292900873280, −19.287152224841314840543083940170, −18.25962046623538113176709842889, −16.53075413477629691194749381772, −15.926386389796745952815047329457, −14.42369050215031008334127985870, −13.15487354511426607523447458160, −11.62371004841410996737189913169, −10.88598224730851055493320620102, −9.61508307234554871027330865379, −8.50978659001541655773405521837, −7.59382256462007068138722594002, −4.964374395328413737099253672325, −3.71244866499038340168992079995, −2.72452133476376661598929120519, −0.28260584564158986588211094826, 1.559797936415798293389634861768, 3.73851423049549854701766582596, 5.54884820719730135150978231629, 7.121946678414538556676219355457, 7.75060931159728590413115958499, 8.780682881010945420974591051691, 10.24841209457702708946694390714, 11.946435219617913186797845287670, 13.09811138997169159113567350779, 14.50156436847091193906149335144, 15.23251280444317794065755214749, 16.46290810538119408109006738056, 17.79447674331118771986307458603, 18.6734647548246072443286176534, 19.57350742406606131843201421219, 20.48145647632439969729317262515, 22.62970411534342363487328803005, 23.6077628921590562442125734591, 24.12690946909375624690652152805, 25.465802347702119438895126214148, 26.06356683576215650223864371968, 27.28520215526350225106868512051, 28.2189449479764107579862352985, 29.44324979121013977019338662045, 30.923573254233670365427807133122

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