Properties

Degree 1
Conductor 149
Sign $0.0392 - 0.999i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.524 − 0.851i)2-s + (0.778 − 0.628i)3-s + (−0.450 − 0.892i)4-s + (0.660 + 0.750i)5-s + (−0.127 − 0.991i)6-s + (0.985 − 0.169i)7-s + (−0.996 − 0.0848i)8-s + (0.210 − 0.977i)9-s + (0.985 − 0.169i)10-s + (−0.450 + 0.892i)11-s + (−0.911 − 0.411i)12-s + (−0.828 + 0.559i)13-s + (0.372 − 0.927i)14-s + (0.985 + 0.169i)15-s + (−0.594 + 0.803i)16-s + (0.778 − 0.628i)17-s + ⋯
L(s,χ)  = 1  + (0.524 − 0.851i)2-s + (0.778 − 0.628i)3-s + (−0.450 − 0.892i)4-s + (0.660 + 0.750i)5-s + (−0.127 − 0.991i)6-s + (0.985 − 0.169i)7-s + (−0.996 − 0.0848i)8-s + (0.210 − 0.977i)9-s + (0.985 − 0.169i)10-s + (−0.450 + 0.892i)11-s + (−0.911 − 0.411i)12-s + (−0.828 + 0.559i)13-s + (0.372 − 0.927i)14-s + (0.985 + 0.169i)15-s + (−0.594 + 0.803i)16-s + (0.778 − 0.628i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.0392 - 0.999i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.0392 - 0.999i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(149\)
\( \varepsilon \)  =  $0.0392 - 0.999i$
motivic weight  =  \(0\)
character  :  $\chi_{149} (17, \cdot )$
Sato-Tate  :  $\mu(37)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 149,\ (0:\ ),\ 0.0392 - 0.999i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(1.362749677 - 1.310254914i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(1.362749677 - 1.310254914i\)
\(L(\chi,1)\)  \(\approx\)  \(1.440399802 - 0.9251088592i\)
\(L(1,\chi)\)  \(\approx\)  \(1.440399802 - 0.9251088592i\)

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

−27.88436584438919594722802400512, −27.31036136054149583443068942413, −26.102694521310480192777792196837, −25.4375467235823325462858008010, −24.336224864097644462784304839592, −23.980047494568404213231935393585, −22.19066974972421906245602455061, −21.36709225615381343223156946021, −20.95941419665529564938031966667, −19.623408717073975710039712481364, −18.05358097960290514080116902915, −17.06527770712901054180161795322, −16.20272439034345951671550448914, −15.08647141161162259230267379702, −14.31206823062413950958874012105, −13.42064668672243222824459149541, −12.40606562698579606000601620724, −10.71745461134995312808677676394, −9.30030494032244209181365349760, −8.411652817713915844954781361872, −7.64153703496236555199467625328, −5.62805627732993147045745753503, −5.04535577055983117931992047358, −3.750059711025585479478774279271, −2.23498418198261615292332676483, 1.820713459150492373171460448425, 2.36996694051358080614289584684, 3.887494486117323360460315557629, 5.28582907725725865066229413791, 6.75767307310485515385601117, 7.92493584637021039430526350083, 9.49195825638520652945400831766, 10.25954511773306258794695955882, 11.62482191855566707288510777619, 12.594780247489140045179171945971, 13.722836432078060282446513409839, 14.54540544337440883173274155588, 14.9508987245712176913067588425, 17.28109701952858561186970462814, 18.389901068567531646099908860927, 18.78670597942707651166703923946, 20.22229837518521537358491287784, 20.822561647714625403955103618, 21.74690198891681338677574312850, 22.94841199164628319061795833131, 23.88757539053309600487729246734, 24.81169379794003340910760377199, 25.92489459814048615368657671569, 26.91202904239066931294962278042, 28.01628587057552549254125988369

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