Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $0.0840 + 0.996i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.342 + 0.939i)2-s + (0.396 + 0.918i)3-s + (−0.766 − 0.642i)4-s + (−0.973 + 0.230i)5-s + (−0.998 + 0.0581i)6-s + (−0.286 + 0.957i)7-s + (0.866 − 0.5i)8-s + (−0.686 + 0.727i)9-s + (0.116 − 0.993i)10-s + (0.116 + 0.993i)11-s + (0.286 − 0.957i)12-s + (−0.957 − 0.286i)13-s + (−0.802 − 0.597i)14-s + (−0.597 − 0.802i)15-s + (0.173 + 0.984i)16-s + (0.984 − 0.173i)17-s + ⋯
L(s,χ)  = 1  + (−0.342 + 0.939i)2-s + (0.396 + 0.918i)3-s + (−0.766 − 0.642i)4-s + (−0.973 + 0.230i)5-s + (−0.998 + 0.0581i)6-s + (−0.286 + 0.957i)7-s + (0.866 − 0.5i)8-s + (−0.686 + 0.727i)9-s + (0.116 − 0.993i)10-s + (0.116 + 0.993i)11-s + (0.286 − 0.957i)12-s + (−0.957 − 0.286i)13-s + (−0.802 − 0.597i)14-s + (−0.597 − 0.802i)15-s + (0.173 + 0.984i)16-s + (0.984 − 0.173i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.0840 + 0.996i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.0840 + 0.996i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $0.0840 + 0.996i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (418, \cdot )$
Sato-Tate  :  $\mu(108)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (1:\ ),\ 0.0840 + 0.996i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.3856741125 + 0.3545136324i$
$L(\frac12,\chi)$  $\approx$  $0.3856741125 + 0.3545136324i$
$L(\chi,1)$  $\approx$  0.3677228431 + 0.5575330892i
$L(1,\chi)$  $\approx$  0.3677228431 + 0.5575330892i

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

−18.60641750778149299190623799387, −17.37806342350742571846538310425, −16.93672354323552067488490658591, −16.52154460430544596385267412484, −15.33218573896635457813611068209, −14.340545816838833641760296004408, −14.02251658873095635414338207386, −13.10064755156807463571691090323, −12.56848190905705367500509886505, −12.07704290101465948469032078213, −11.285255497309703914137226076953, −10.72287044885360807354952680627, −9.872965499883971182661205482427, −8.968764142124591836220023997601, −8.429546565693504050085085384868, −7.710777606883335663536068940883, −7.27194796030506929223522164238, −6.382249725548665137800994980883, −5.23086746317792231281421763814, −4.096638186179358268106285318711, −3.802388837783838890949392284453, −2.865879329919585141006824163798, −2.20938046032287904627634447751, −0.96802096826248608505098309522, −0.6005415310474831138032748044, 0.15102177849202397642217789351, 1.65543470350611826360297612374, 2.77507330555227377728288123100, 3.50152494787518660473784768526, 4.39583258579754659376521153640, 5.02957498374582448819605204671, 5.58008249274164103286073452904, 6.68038234490330606185882839470, 7.39327271538255033485030069668, 8.00779426463684345052863752966, 8.71128679004379696369185026642, 9.34933184902957045854049649389, 9.99066996464168987616873193951, 10.567049869517896097896195666784, 11.55475468799683416281236246630, 12.371056767972060238600078313090, 12.94838844788674089390839177649, 14.214492046454328352747995061346, 14.779238396188014540781561151047, 15.07271844049075880942440045158, 15.60220229431521017095223231643, 16.3915101833127263491247454872, 16.77583164774569006064008919768, 17.7487898982465148800066323920, 18.37869168957726166046988005976

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