Properties

Degree 1
Conductor 311
Sign $0.986 - 0.162i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.874 + 0.485i)2-s + (−0.874 − 0.485i)3-s + (0.528 − 0.848i)4-s + (−0.612 − 0.790i)5-s + 6-s + (0.979 + 0.201i)7-s + (−0.0506 + 0.998i)8-s + (0.528 + 0.848i)9-s + (0.918 + 0.394i)10-s + (−0.250 − 0.968i)11-s + (−0.874 + 0.485i)12-s + (0.528 + 0.848i)13-s + (−0.954 + 0.299i)14-s + (0.151 + 0.988i)15-s + (−0.440 − 0.897i)16-s + (0.918 + 0.394i)17-s + ⋯
L(s,χ)  = 1  + (−0.874 + 0.485i)2-s + (−0.874 − 0.485i)3-s + (0.528 − 0.848i)4-s + (−0.612 − 0.790i)5-s + 6-s + (0.979 + 0.201i)7-s + (−0.0506 + 0.998i)8-s + (0.528 + 0.848i)9-s + (0.918 + 0.394i)10-s + (−0.250 − 0.968i)11-s + (−0.874 + 0.485i)12-s + (0.528 + 0.848i)13-s + (−0.954 + 0.299i)14-s + (0.151 + 0.988i)15-s + (−0.440 − 0.897i)16-s + (0.918 + 0.394i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.986 - 0.162i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.986 - 0.162i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(311\)
\( \varepsilon \)  =  $0.986 - 0.162i$
motivic weight  =  \(0\)
character  :  $\chi_{311} (113, \cdot )$
Sato-Tate  :  $\mu(31)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 311,\ (0:\ ),\ 0.986 - 0.162i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.6186060869 - 0.05062477132i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.6186060869 - 0.05062477132i\)
\(L(\chi,1)\)  \(\approx\)  \(0.5928203038 + 0.02259234308i\)
\(L(1,\chi)\)  \(\approx\)  \(0.5928203038 + 0.02259234308i\)

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

−25.63072261329425465758778584548, −24.34402799471203097336006199822, −23.21020498957877935363547751465, −22.63809179888528357941247019532, −21.51892719938497883705312818727, −20.74415563725029392908363027157, −19.98851952599735641309688161541, −18.63026734602465242230528724383, −18.05238120603108269818253920053, −17.39079621614327365573141073485, −16.346222452025573483217849189926, −15.385133220044074844555897692008, −14.70071720764379850040616418337, −12.85892111421441196862695584677, −11.94614275040896794151287477756, −11.08857644730511248393148966833, −10.58261124722109595683006364780, −9.69452412341754413787130273268, −8.24363659813419733616147152561, −7.42800855670797695712776520645, −6.437561561226324133085921916281, −4.86837627536814765896425610741, −3.849296971303681392139302328414, −2.549662404225435381371566116716, −0.89548523680608904583434996638, 0.93186175951574247925427953256, 1.84762862825308809445289266721, 4.17293402120177473550360099231, 5.466355377694454399380759046693, 6.01488991870284715523706761415, 7.536877001486298964466633474051, 8.072735738845896090805634622498, 9.00335530452555765464929613435, 10.430421806299514238294427679203, 11.44519504643825311023878236405, 11.81033783663406089921167355550, 13.22489995776212967128129890657, 14.392895989601484483611205412254, 15.5710425934905701474229082419, 16.49336233445674308401527413451, 16.920920644577130998869397058434, 17.93131369265207745906326425476, 18.92662187930523004842060879705, 19.27215457029683111709099186385, 20.82167236960674413949408675169, 21.36543130953501158659454369716, 23.08636551265585284348407455969, 23.73431710608568859934350826234, 24.25082264692730684742595006471, 24.962275296179604183685198361394

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