Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $-0.997 + 0.0665i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.5 − 0.866i)2-s + (−0.993 + 0.116i)3-s + (−0.5 − 0.866i)4-s + (0.686 − 0.727i)5-s + (−0.396 + 0.918i)6-s + (0.973 + 0.230i)7-s − 8-s + (0.973 − 0.230i)9-s + (−0.286 − 0.957i)10-s + (−0.286 + 0.957i)11-s + (0.597 + 0.802i)12-s + (0.686 − 0.727i)13-s + (0.686 − 0.727i)14-s + (−0.597 + 0.802i)15-s + (−0.5 + 0.866i)16-s − 17-s + ⋯
L(s,χ)  = 1  + (0.5 − 0.866i)2-s + (−0.993 + 0.116i)3-s + (−0.5 − 0.866i)4-s + (0.686 − 0.727i)5-s + (−0.396 + 0.918i)6-s + (0.973 + 0.230i)7-s − 8-s + (0.973 − 0.230i)9-s + (−0.286 − 0.957i)10-s + (−0.286 + 0.957i)11-s + (0.597 + 0.802i)12-s + (0.686 − 0.727i)13-s + (0.686 − 0.727i)14-s + (−0.597 + 0.802i)15-s + (−0.5 + 0.866i)16-s − 17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.997 + 0.0665i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.997 + 0.0665i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $-0.997 + 0.0665i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (2805, \cdot )$
Sato-Tate  :  $\mu(54)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ -0.997 + 0.0665i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.05192349123 - 1.558131659i$
$L(\frac12,\chi)$  $\approx$  $-0.05192349123 - 1.558131659i$
$L(\chi,1)$  $\approx$  0.8267435782 - 0.7580827441i
$L(1,\chi)$  $\approx$  0.8267435782 - 0.7580827441i

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

−18.54975585261915200812767646236, −17.911662178641812080837468795074, −17.38069095615131905793524098086, −16.91569982303206550044132037460, −15.96997150388156470871884265314, −15.65485174510640461351098688194, −14.635424496452317844440157771690, −13.89463391314623873828050666678, −13.68698358128576729066443625611, −12.83046394586793432563193819944, −11.88054132585169300735335134742, −11.29583104111734513774290423131, −10.75847879290364979572181245795, −9.95353762727281395960744730007, −8.88591266912434570214266596695, −8.238360049021195740911074453512, −7.365310286509857633551059998033, −6.70073675935518276122928069420, −6.1300918684339597580912164146, −5.55241134252871824156870453311, −4.84449970679324315520729761444, −4.056384785653190285223587198883, −3.25197369533748282433075173954, −2.04734047429707068515510445421, −1.21386674054114285910927792359, 0.44284670117055980197464142491, 1.284003540441430560882707454148, 2.04531499901544009436663096631, 2.733413082330090944028714502901, 4.29688853657655352191559819700, 4.61168983726353629272424354932, 5.04665078342545633264600470238, 6.04828108014680684600607142317, 6.35656491490035087104424901888, 7.66128107613603022011569219601, 8.64974245613366117305555970953, 9.246683987940771732867883259177, 10.139199084648628449438854032317, 10.63620415422377740958192383640, 11.2272983102255182199116547609, 12.078544797294411145511553383423, 12.45744764915126160182405557764, 13.30068403286261001408379475164, 13.62321381440284336231175437489, 14.733161728440476699406110918403, 15.476940513903534828710317673622, 15.82433656975510126429622866705, 17.26170434334581355509767392964, 17.41636381911524483787446739231, 18.11230402361294453416454128597

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