Properties

Degree $1$
Conductor $311$
Sign $0.945 - 0.326i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.758 − 0.651i)2-s + (−0.758 + 0.651i)3-s + (0.151 + 0.988i)4-s + (0.820 + 0.571i)5-s + 6-s + (−0.612 + 0.790i)7-s + (0.528 − 0.848i)8-s + (0.151 − 0.988i)9-s + (−0.250 − 0.968i)10-s + (0.347 − 0.937i)11-s + (−0.758 − 0.651i)12-s + (0.151 − 0.988i)13-s + (0.979 − 0.201i)14-s + (−0.994 + 0.101i)15-s + (−0.954 + 0.299i)16-s + (−0.250 − 0.968i)17-s + ⋯
L(s,χ)  = 1  + (−0.758 − 0.651i)2-s + (−0.758 + 0.651i)3-s + (0.151 + 0.988i)4-s + (0.820 + 0.571i)5-s + 6-s + (−0.612 + 0.790i)7-s + (0.528 − 0.848i)8-s + (0.151 − 0.988i)9-s + (−0.250 − 0.968i)10-s + (0.347 − 0.937i)11-s + (−0.758 − 0.651i)12-s + (0.151 − 0.988i)13-s + (0.979 − 0.201i)14-s + (−0.994 + 0.101i)15-s + (−0.954 + 0.299i)16-s + (−0.250 − 0.968i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.945 - 0.326i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.945 - 0.326i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(311\)
Sign: $0.945 - 0.326i$
Motivic weight: \(0\)
Character: $\chi_{311} (91, \cdot )$
Sato-Tate group: $\mu(31)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 311,\ (0:\ ),\ 0.945 - 0.326i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(0.7199443969 - 0.1206862181i\)
\(L(\frac12,\chi)\) \(\approx\) \(0.7199443969 - 0.1206862181i\)
\(L(\chi,1)\) \(\approx\) \(0.6795932439 - 0.04298389292i\)
\(L(1,\chi)\) \(\approx\) \(0.6795932439 - 0.04298389292i\)

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

−25.353249712140922355497925410175, −24.42964975836798073706106070295, −23.673392627176458854900313386229, −22.96802633849536753651712372033, −21.921400156544840213009094223504, −20.55352811885161942386029210771, −19.65619679434170147087176629170, −18.813847406733282227912134082052, −17.77395486875386614855991561088, −17.0970158432783799584212435605, −16.673849625448332594691627418222, −15.63017033924322816368505165830, −14.15275985527683281457870647687, −13.46862779225039513793225118895, −12.42979660938507759895400429067, −11.24710697636693315269204200624, −10.07562625740374356474362077762, −9.53587496025611309852333740033, −8.21144112990133996632951505165, −7.04183298565100412390228043446, −6.473774966902552855277511668183, −5.48462678322724061965963129322, −4.36237686460336751124648719954, −1.93475801536947595425068757393, −1.13803836503766234763089671661, 0.83554400744856216639503666027, 2.731528014513371600314691661814, 3.31778365581601187421884203016, 5.07992247770597016061041955807, 6.12343117814222620510673715056, 7.04226816700059168964316917158, 8.811856065185448658377548894845, 9.36609931161660927008317763565, 10.38859153973540976392787839853, 11.00831629112162753221015562442, 12.00738823599710440828188315704, 12.935073727338244071119170080589, 14.137079671661268707213940702200, 15.63539503478265690732371714017, 16.16843624199507107985377806193, 17.28008234408864620024833705548, 18.04275731411322064580611005497, 18.616258417671982342289669279097, 19.792213697689376065684596888740, 20.883674341591573237283516430325, 21.66757111447926549573648334487, 22.28010671592782030637199516733, 22.84088019427707717710906024026, 24.70723292823970392892456778377, 25.28877334995633722393076325621

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