Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $0.674 - 0.738i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.674 - 0.738i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.674 - 0.738i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $0.674 - 0.738i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (942, \cdot )$
Sato-Tate  :  $\mu(108)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ 0.674 - 0.738i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.1060192739 - 0.04675699496i$
$L(\frac12,\chi)$  $\approx$  $0.1060192739 - 0.04675699496i$
$L(\chi,1)$  $\approx$  0.4186971181 + 0.2267128365i
$L(1,\chi)$  $\approx$  0.4186971181 + 0.2267128365i

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

−18.41124603344713725626907085492, −17.691998594428804083679247997249, −17.20122017946624869469771624896, −16.90781265821492263638725970079, −16.13353845744662261253252829897, −15.497651478500735251472141937593, −14.448487518475637105381006389615, −13.52923253659704002726803879445, −13.03197921491885636044621823201, −12.14022446571476112566932039909, −11.82102994486084592447876312037, −11.009759694722557475655033391933, −9.9709646969025736989221245813, −9.73477231854574580307926315981, −8.91052102305044867939882038198, −7.86867192444795597131705642085, −7.32495679313088978901842906793, −7.04468079580084000300779114501, −5.983728728516089575126206268176, −5.24679061020813361811132535311, −4.53607757313707178515615513755, −3.31232517063126190969277336685, −2.17341193453370836728640621938, −1.59272991327123119978281260187, −0.7669093667208950496599245861, 0.065013336199157292065896152609, 1.45891792440689371325134903156, 2.619199454628344325939280780181, 2.98884281784296751011295339293, 3.91763758783626765910311398807, 5.18655242455157240700392285554, 5.8301092846910801017529036870, 6.49346868349900977252771096938, 6.96518372075781089738532168818, 8.11313851227696740602450882429, 8.85063535911822355528225192832, 9.48172948225201704749041612869, 10.0150161501969949554330577981, 10.80044319128091260295087652799, 11.23172042537432168132752327065, 11.941882696614920120512236117444, 12.50652768002838788405735050240, 13.79779460839412282160878713520, 14.7020919127031223366645107950, 15.16179108046341131686150814503, 15.69144857280189316486092858325, 16.506732514451433838938867052512, 17.01264948465623779019361016925, 17.69268219243794397011858111752, 18.42818231177726816647380652562

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