Properties

Degree 1
Conductor 19
Sign $-0.305 + 0.952i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + 7-s + 8-s + (−0.5 − 0.866i)9-s + (−0.5 − 0.866i)10-s + 11-s + 12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯
L(s,χ)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + 7-s + 8-s + (−0.5 − 0.866i)9-s + (−0.5 − 0.866i)10-s + 11-s + 12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.305 + 0.952i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 19 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.305 + 0.952i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(19\)
\( \varepsilon \)  =  $-0.305 + 0.952i$
motivic weight  =  \(0\)
character  :  $\chi_{19} (7, \cdot )$
Sato-Tate  :  $\mu(3)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 19,\ (0:\ ),\ -0.305 + 0.952i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.2653973789 + 0.3639800608i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.2653973789 + 0.3639800608i\)
\(L(\chi,1)\)  \(\approx\)  \(0.4992035152 + 0.4022141313i\)
\(L(1,\chi)\)  \(\approx\)  \(0.4992035152 + 0.4022141313i\)

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

−40.22787418304894314564986293329, −39.069391278677512493842139846169, −37.379802909321561811105896280132, −36.10464510822524108850693114442, −35.416076191824706112222999318218, −33.93294256575851837093282442897, −31.60225228856004011900831156716, −30.57284043815652044178886263982, −29.31727491172878771802277642022, −28.09764486782685251781474558821, −27.15357072471301856669666842041, −24.93890194818570857552811786818, −23.73867106697467176894410671072, −22.02787851228323272114484102839, −20.35605651057973071520226704261, −19.21004735047649087413862767539, −17.714719249995459862616200701135, −16.6734732907445739545554694964, −13.79080829985181288954638451241, −12.0908007615320473227409179149, −11.428432959283788437662983956906, −8.97228608572294362591041132672, −7.498422600723132635590660250524, −4.639645630545534199223837216, −1.57320808455217667944576519897, 4.40883445095759858523275964106, 6.29765489642468131575353719258, 8.1872505424540154019873653341, 10.10796142252987842080819448827, 11.396325399764076250182529561711, 14.57003800972870037766640785250, 15.255028388258225412799133289879, 16.97134418412441583103620362119, 18.04739280159696783196787643803, 19.86600550830137009502699607557, 22.003361277896269211601619608589, 23.084835327948264935378218525980, 24.577869497894815801446247298544, 26.39821209218555236472220813456, 27.2216007043267593171072508725, 28.17673136074784985593507284243, 30.2828022077364862616451162762, 32.089972383249371794155173998363, 33.353731970792871474749548817303, 34.32332760220468950085327732698, 35.182945754836544710002587601690, 37.23797040928228191804280146842, 38.042880905136483901315133727015, 39.72713785644655001036063400490, 41.1253901054614516133941262833

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