Properties

Degree 1
Conductor 151
Sign $0.996 - 0.0838i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.978 − 0.207i)2-s + (−0.637 + 0.770i)3-s + (0.913 + 0.406i)4-s + (0.944 + 0.328i)5-s + (0.783 − 0.621i)6-s + (0.604 − 0.796i)7-s + (−0.809 − 0.587i)8-s + (−0.187 − 0.982i)9-s + (−0.855 − 0.518i)10-s + (0.985 − 0.166i)11-s + (−0.895 + 0.444i)12-s + (−0.699 − 0.714i)13-s + (−0.756 + 0.653i)14-s + (−0.855 + 0.518i)15-s + (0.669 + 0.743i)16-s + (0.387 − 0.921i)17-s + ⋯
L(s,χ)  = 1  + (−0.978 − 0.207i)2-s + (−0.637 + 0.770i)3-s + (0.913 + 0.406i)4-s + (0.944 + 0.328i)5-s + (0.783 − 0.621i)6-s + (0.604 − 0.796i)7-s + (−0.809 − 0.587i)8-s + (−0.187 − 0.982i)9-s + (−0.855 − 0.518i)10-s + (0.985 − 0.166i)11-s + (−0.895 + 0.444i)12-s + (−0.699 − 0.714i)13-s + (−0.756 + 0.653i)14-s + (−0.855 + 0.518i)15-s + (0.669 + 0.743i)16-s + (0.387 − 0.921i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.996 - 0.0838i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.996 - 0.0838i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(151\)
\( \varepsilon \)  =  $0.996 - 0.0838i$
motivic weight  =  \(0\)
character  :  $\chi_{151} (49, \cdot )$
Sato-Tate  :  $\mu(75)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 151,\ (0:\ ),\ 0.996 - 0.0838i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.7393071006 + 0.03103948158i$
$L(\frac12,\chi)$  $\approx$  $0.7393071006 + 0.03103948158i$
$L(\chi,1)$  $\approx$  0.7330231120 + 0.01592743239i
$L(1,\chi)$  $\approx$  0.7330231120 + 0.01592743239i

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

−28.140438928498751202851643657633, −27.42861643490167193240305034646, −25.85955278238954575781126022549, −25.1165110622826325502755507655, −24.40767502016488690410927278040, −23.64847029611542722431994351635, −21.91966820673271807288227598257, −21.30539351439634297998439081344, −19.710398613642354480865738963583, −19.000785516520864869742193993207, −17.81831041540139539529250808204, −17.336503982154221931714830970531, −16.57801100026785522666391492173, −15.02969693785991703609483194970, −13.98877494219034086231640122141, −12.419229443084017030851894924059, −11.716877073142048226093895792110, −10.50014441110881233404633315421, −9.25057644539613271671333092782, −8.3698936849143550267256821278, −6.95294624574542290706576436092, −6.10043071367410935370372342077, −5.06032787011329234078802073198, −2.19015374356054166970417664603, −1.48049172988102795669434405447, 1.103265776449733351133391809457, 2.84903611025673977837801405221, 4.44824396739024559067804362126, 5.9882189615443533025890450012, 6.96780049507282624247595612732, 8.471346224643072758931220702726, 9.816905084331866053868220227863, 10.26203609018235992990675072763, 11.29161412774195541873408580197, 12.329201565374632116596078772650, 14.11351521600848862327523141434, 15.06045319537558196488582348632, 16.49373137395846128619193830071, 17.224426653585462449413633964454, 17.689327159152595988989222154017, 18.95587173117019316557852714609, 20.38349292715399665784631929673, 20.93998891550204275751800656259, 21.98634098899238916744126467913, 22.89555219769677896953243130965, 24.45458064277090627780564758836, 25.277559599285595325260828059146, 26.48430166603185889053010742278, 27.143535981840370575424551922550, 27.809382563562976092234273668382

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