Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $-0.983 + 0.183i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.766 − 0.642i)2-s + (0.686 + 0.727i)3-s + (0.173 − 0.984i)4-s + (0.448 − 0.893i)5-s + (0.993 + 0.116i)6-s + (−0.835 + 0.549i)7-s + (−0.5 − 0.866i)8-s + (−0.0581 + 0.998i)9-s + (−0.230 − 0.973i)10-s + (0.230 − 0.973i)11-s + (0.835 − 0.549i)12-s + (−0.835 + 0.549i)13-s + (−0.286 + 0.957i)14-s + (0.957 − 0.286i)15-s + (−0.939 − 0.342i)16-s + (0.939 + 0.342i)17-s + ⋯
L(s,χ)  = 1  + (0.766 − 0.642i)2-s + (0.686 + 0.727i)3-s + (0.173 − 0.984i)4-s + (0.448 − 0.893i)5-s + (0.993 + 0.116i)6-s + (−0.835 + 0.549i)7-s + (−0.5 − 0.866i)8-s + (−0.0581 + 0.998i)9-s + (−0.230 − 0.973i)10-s + (0.230 − 0.973i)11-s + (0.835 − 0.549i)12-s + (−0.835 + 0.549i)13-s + (−0.286 + 0.957i)14-s + (0.957 − 0.286i)15-s + (−0.939 − 0.342i)16-s + (0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.983 + 0.183i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.983 + 0.183i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $-0.983 + 0.183i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (1842, \cdot )$
Sato-Tate  :  $\mu(108)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ -0.983 + 0.183i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.1050153949 - 1.136454258i$
$L(\frac12,\chi)$  $\approx$  $-0.1050153949 - 1.136454258i$
$L(\chi,1)$  $\approx$  1.363234765 - 0.5982206425i
$L(1,\chi)$  $\approx$  1.363234765 - 0.5982206425i

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

−18.95965433896051527274163516299, −17.92842505920951143359249325772, −17.44622042075042985690827135258, −16.943713649330458016232642816904, −15.860580696284167271701808360540, −15.205492181159318612863279399610, −14.51428431859637732686664545546, −14.27686069676456424995915456357, −13.33240348231874602626118268701, −12.988324052171945843493157544704, −12.19914791241965852914601396439, −11.67213531244641301437932056134, −10.33974720041239430916613978455, −9.83414438544980250883880145874, −9.12393812071107955053178881854, −7.85568272445303243682392248408, −7.545595755365410897834059755371, −6.96872447713999775096601536125, −6.20395700110513723828298258601, −5.77675967526791953925836514324, −4.48347829736794929047382536043, −3.75956501083250077278787860330, −2.9887500227846437701375585969, −2.48968215909745561861482257002, −1.57147557057651001188238465422, 0.18237055008748813605860655490, 1.584733053680445646473078628400, 2.26831499343034282937464426250, 3.07519593012189824380309447355, 3.68553129105300161120802094480, 4.58479937068956415550380103416, 5.06309572194482534113169711978, 6.003702285421674202692765478448, 6.41330088893664838276277566917, 7.811282800456509503941069242701, 8.76336096740323487607567023798, 9.14093877312641563597643944507, 9.91407558885028082856831294567, 10.31255681841372833035040295034, 11.29309725857914454051427989240, 12.14229118273569074528819551234, 12.65865667015552638960056997982, 13.33970597980969068157288065622, 13.994258022757574812217607771831, 14.56469074018172371836394763551, 15.23877611195155479604223195919, 16.0938936730169992678828250140, 16.48394349366710948361467514443, 17.14357974665698765187536384733, 18.51402743636208922770928746019

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