Properties

Degree 1
Conductor 31
Sign $-0.107 - 0.994i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.309 − 0.951i)2-s + (0.978 + 0.207i)3-s + (−0.809 − 0.587i)4-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)6-s + (0.913 − 0.406i)7-s + (−0.809 + 0.587i)8-s + (0.913 + 0.406i)9-s + (−0.978 + 0.207i)10-s + (0.104 − 0.994i)11-s + (−0.669 − 0.743i)12-s + (−0.669 + 0.743i)13-s + (−0.104 − 0.994i)14-s + (−0.309 − 0.951i)15-s + (0.309 + 0.951i)16-s + (0.104 + 0.994i)17-s + ⋯
L(s,χ)  = 1  + (0.309 − 0.951i)2-s + (0.978 + 0.207i)3-s + (−0.809 − 0.587i)4-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)6-s + (0.913 − 0.406i)7-s + (−0.809 + 0.587i)8-s + (0.913 + 0.406i)9-s + (−0.978 + 0.207i)10-s + (0.104 − 0.994i)11-s + (−0.669 − 0.743i)12-s + (−0.669 + 0.743i)13-s + (−0.104 − 0.994i)14-s + (−0.309 − 0.951i)15-s + (0.309 + 0.951i)16-s + (0.104 + 0.994i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.107 - 0.994i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.107 - 0.994i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(31\)
\( \varepsilon \)  =  $-0.107 - 0.994i$
motivic weight  =  \(0\)
character  :  $\chi_{31} (3, \cdot )$
Sato-Tate  :  $\mu(30)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 31,\ (1:\ ),\ -0.107 - 0.994i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(1.304781025 - 1.453390150i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(1.304781025 - 1.453390150i\)
\(L(\chi,1)\)  \(\approx\)  \(1.254456837 - 0.8493788121i\)
\(L(1,\chi)\)  \(\approx\)  \(1.254456837 - 0.8493788121i\)

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

−36.7352267444871806213994763291, −35.45795780563019512202741929504, −34.3514405366979757013803867354, −33.25775512083228378408567920330, −31.77909976775941453752177270690, −30.911265799675803606633266089309, −30.153637580698840516558496302061, −27.49762137878637496753781405449, −26.701389680363194477110150218751, −25.410716138682208671339472950909, −24.55076704375468420986431922134, −23.161952712965905434709932129386, −21.86301312217391236831157986110, −20.29124600973329709860846246129, −18.62253482082533926580340639660, −17.655638330697646485805253450889, −15.347502664242961231267668873906, −14.94060758201728646202150162162, −13.64103968058234846950207462326, −11.95931418621944453693605239813, −9.55664055691256227584424367710, −7.8978362479925052067343437037, −7.11116185129586095376766758412, −4.77719687707376538197137072083, −2.91657310270090614760471261710, 1.50350285479840071003541054297, 3.635717547566635157760076031663, 4.880307193502959789151666618, 8.06883272743676520420270812263, 9.16639843019713844172821068560, 10.867625060069604766756803370832, 12.380027976802252751466650911, 13.79424788750953437974391466630, 14.82125721465484063320824846401, 16.724454478781870309796007196746, 18.76875194715645248693190470956, 19.82726120932230546791467489685, 20.80309616338364128570953013543, 21.71077412139138055803492672327, 23.754724214080354983608787171452, 24.52920217336992475685017480029, 26.762596156853940845579822262638, 27.34418363170132115978691325907, 28.84059828891699174987511478409, 30.29481819231005276612293972889, 31.2731984185502489767414394044, 32.10347356525466148680924644738, 33.22224004095439253005742255749, 35.42117547996811157641656938200, 36.80627562924451668192883355138

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