Properties

Degree $1$
Conductor $311$
Sign $0.335 + 0.942i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.820 + 0.571i)2-s + (0.820 − 0.571i)3-s + (0.347 + 0.937i)4-s + (0.151 + 0.988i)5-s + 6-s + (0.528 + 0.848i)7-s + (−0.250 + 0.968i)8-s + (0.347 − 0.937i)9-s + (−0.440 + 0.897i)10-s + (−0.954 − 0.299i)11-s + (0.820 + 0.571i)12-s + (0.347 − 0.937i)13-s + (−0.0506 + 0.998i)14-s + (0.688 + 0.724i)15-s + (−0.758 + 0.651i)16-s + (−0.440 + 0.897i)17-s + ⋯
L(s,χ)  = 1  + (0.820 + 0.571i)2-s + (0.820 − 0.571i)3-s + (0.347 + 0.937i)4-s + (0.151 + 0.988i)5-s + 6-s + (0.528 + 0.848i)7-s + (−0.250 + 0.968i)8-s + (0.347 − 0.937i)9-s + (−0.440 + 0.897i)10-s + (−0.954 − 0.299i)11-s + (0.820 + 0.571i)12-s + (0.347 − 0.937i)13-s + (−0.0506 + 0.998i)14-s + (0.688 + 0.724i)15-s + (−0.758 + 0.651i)16-s + (−0.440 + 0.897i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.335 + 0.942i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.335 + 0.942i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(311\)
Sign: $0.335 + 0.942i$
Motivic weight: \(0\)
Character: $\chi_{311} (225, \cdot )$
Sato-Tate group: $\mu(31)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 311,\ (0:\ ),\ 0.335 + 0.942i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(2.147783208 + 1.515659177i\)
\(L(\frac12,\chi)\) \(\approx\) \(2.147783208 + 1.515659177i\)
\(L(\chi,1)\) \(\approx\) \(1.903963796 + 0.8195696933i\)
\(L(1,\chi)\) \(\approx\) \(1.903963796 + 0.8195696933i\)

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

−24.877682587822726386595479633698, −24.06205250158728653726686322733, −23.37630128318002743011814555691, −22.18895844044946753616558823410, −21.131408427862919393531891290415, −20.59936853778726383022464700593, −20.27612092226261766609103485532, −19.124460749700535962049918582006, −18.02926424909886950242840922674, −16.31433631550156537328290497588, −16.10275603961749356358374715875, −14.6754674835115627197330847055, −13.98827752048259640417188099754, −13.28425274719865312091136842188, −12.33859708176354973894165433711, −11.06585242623671422363834518338, −10.23296469215956097999317291649, −9.311688303200023603447368744, −8.24065779342826345071417036705, −7.036473393325897580134291295938, −5.33874960017916769989473460535, −4.58446222648653360765285522906, −3.874680539283172957243114285443, −2.45123395910553069513032302963, −1.390339496921960078519999254696, 2.22458803333496172283309980624, 2.80534254383040509889770983557, 3.9357396107849557741327687265, 5.56575396642185764084449879774, 6.24038653207978065604475348448, 7.58772685799130297036317008698, 8.00343643643257322181107898458, 9.19919093918065416258527898970, 10.8025458713825708989308399926, 11.709394606877824638573147906977, 13.01564511736500461716436446465, 13.428055240767027511316059391407, 14.5729978180506923551910845277, 15.233506580553678227887127631, 15.70936961533490820441726148658, 17.66303087254903650038502562776, 17.97391012345584139596973058289, 19.06384810651538194669217380979, 20.14851099090435575702860836618, 21.294836305651444349304265899656, 21.70864167535609573656091805022, 22.88950127842405775144351563196, 23.75839001219569318398893218781, 24.53992900344014555763587194800, 25.318307250559564761903048386

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