Properties

Degree 1
Conductor 37
Sign $-0.665 - 0.746i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.173 − 0.984i)2-s + (0.173 − 0.984i)3-s + (−0.939 + 0.342i)4-s + (−0.766 − 0.642i)5-s − 6-s + (0.766 + 0.642i)7-s + (0.5 + 0.866i)8-s + (−0.939 − 0.342i)9-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (0.173 + 0.984i)12-s + (0.939 − 0.342i)13-s + (0.5 − 0.866i)14-s + (−0.766 + 0.642i)15-s + (0.766 − 0.642i)16-s + (0.939 + 0.342i)17-s + ⋯
L(s,χ)  = 1  + (−0.173 − 0.984i)2-s + (0.173 − 0.984i)3-s + (−0.939 + 0.342i)4-s + (−0.766 − 0.642i)5-s − 6-s + (0.766 + 0.642i)7-s + (0.5 + 0.866i)8-s + (−0.939 − 0.342i)9-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (0.173 + 0.984i)12-s + (0.939 − 0.342i)13-s + (0.5 − 0.866i)14-s + (−0.766 + 0.642i)15-s + (0.766 − 0.642i)16-s + (0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.665 - 0.746i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.665 - 0.746i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(37\)
\( \varepsilon \)  =  $-0.665 - 0.746i$
motivic weight  =  \(0\)
character  :  $\chi_{37} (3, \cdot )$
Sato-Tate  :  $\mu(18)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 37,\ (0:\ ),\ -0.665 - 0.746i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.2728970460 - 0.6091696721i$
$L(\frac12,\chi)$  $\approx$  $0.2728970460 - 0.6091696721i$
$L(\chi,1)$  $\approx$  0.5730256230 - 0.6028616835i
$L(1,\chi)$  $\approx$  0.5730256230 - 0.6028616835i

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

−35.98481981469406901194170491145, −34.450878824919384579074961041155, −33.72370093807847105049095112730, −32.8104936441167454980384843732, −31.42460828509511208582358851621, −30.58374534777128762718162974694, −28.1568692134841556434653762626, −27.35234622302266576426852016860, −26.37811385004794953255824231267, −25.52412406137941537805807082375, −23.557185588307763619515639743372, −23.02220995021267017846663693034, −21.43418170723832982566322134725, −19.97289309798799357463800245099, −18.39327756383791419107886546660, −17.05280211805387660463422622819, −15.7187005940795798310575442379, −14.92992840018631673601381585473, −13.76162252981835000942418011962, −11.24575579035053140899260201945, −9.99471994821601930946177458734, −8.35020742300255258348190221368, −7.15317776955223512678826869163, −5.05529475560601806138394609841, −3.79770706941784267078472872641, 1.36111872145182545173201029804, 3.32290403676842290931058032293, 5.420362327351430206624935192594, 8.10272357707850378095747145648, 8.58139326078342803618355406973, 10.97700886914988791975009604334, 12.09429545145793591479221093531, 13.02738078356333233080063428918, 14.5248929595323202146958025779, 16.63398122301660007436649276387, 18.294280900585872254205705913206, 18.90854624061228455909847907802, 20.2857075846320586014774606501, 21.22532623007380492916163247380, 23.095673111181431982202952528895, 23.99340056835739397546274868979, 25.41089174075348964331022724642, 27.11102290279824701784820907542, 28.12653650299418180341509909885, 29.18124676862525714569439556429, 30.500940037908529412695328711730, 31.22779458708764400247871823403, 32.17589015975588054808924110867, 34.598115521742330046183640873689, 35.3246143635791424308720588999

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