Properties

Degree 1
Conductor 149
Sign $-0.568 + 0.822i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.450 + 0.892i)2-s + (0.210 + 0.977i)3-s + (−0.594 − 0.803i)4-s + (−0.127 − 0.991i)5-s + (−0.967 − 0.251i)6-s + (0.942 + 0.333i)7-s + (0.985 − 0.169i)8-s + (−0.911 + 0.411i)9-s + (0.942 + 0.333i)10-s + (−0.594 + 0.803i)11-s + (0.660 − 0.750i)12-s + (0.372 + 0.927i)13-s + (−0.721 + 0.691i)14-s + (0.942 − 0.333i)15-s + (−0.292 + 0.956i)16-s + (0.210 + 0.977i)17-s + ⋯
L(s,χ)  = 1  + (−0.450 + 0.892i)2-s + (0.210 + 0.977i)3-s + (−0.594 − 0.803i)4-s + (−0.127 − 0.991i)5-s + (−0.967 − 0.251i)6-s + (0.942 + 0.333i)7-s + (0.985 − 0.169i)8-s + (−0.911 + 0.411i)9-s + (0.942 + 0.333i)10-s + (−0.594 + 0.803i)11-s + (0.660 − 0.750i)12-s + (0.372 + 0.927i)13-s + (−0.721 + 0.691i)14-s + (0.942 − 0.333i)15-s + (−0.292 + 0.956i)16-s + (0.210 + 0.977i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.568 + 0.822i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.568 + 0.822i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(149\)
\( \varepsilon \)  =  $-0.568 + 0.822i$
motivic weight  =  \(0\)
character  :  $\chi_{149} (33, \cdot )$
Sato-Tate  :  $\mu(37)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 149,\ (0:\ ),\ -0.568 + 0.822i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.4094946040 + 0.7810591753i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.4094946040 + 0.7810591753i\)
\(L(\chi,1)\)  \(\approx\)  \(0.6742814158 + 0.5649059584i\)
\(L(1,\chi)\)  \(\approx\)  \(0.6742814158 + 0.5649059584i\)

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

−27.706049879316189201342935079662, −26.906065735029319872324808928364, −26.00324343560794319235445263416, −25.02775687591252633732021790887, −23.65648108933629380796888621486, −22.91858058245785406843804298920, −21.68619031825318931860996219541, −20.66671828785624273222660909023, −19.67534961906932169957996692661, −18.795025523311457601170730707360, −17.94787547110623119295292180806, −17.47940783160269271280410228411, −15.64461911041665187340603350493, −14.031740544187692289952081377030, −13.631495458803924209060457078861, −12.2301221010641435167635696492, −11.16859373461845904138566665463, −10.62039461345976541685075123492, −8.891625010455282047386339774769, −7.852982191837672399643080232, −7.12300108197102295430357125047, −5.30656362781146801138337042828, −3.34995739476853462982494678556, −2.53587201102780398193324160833, −0.95563730997590794887549818785, 1.75930169510483242894584004392, 4.27888789023979932651807465425, 4.86638020672221305129700388630, 6.04422818993475953819663914103, 7.98064755433774098404082949899, 8.491030512247811283537145800072, 9.58142651741166737570303407051, 10.58324018256173473174149370949, 12.01360051063029805944430977692, 13.5616896056751616113618318397, 14.725560405887394568474842774462, 15.375987234030167199880173271370, 16.48599466517277748848878417370, 17.07355341647952082165804025526, 18.26952166483402308240987634708, 19.5159440137908940741914417384, 20.72976304719056720328322548989, 21.28091249894951968276381678131, 22.8070195969385540185187286447, 23.730153613089854919045136200, 24.66465666564791426569959950987, 25.592596491710236943036660349566, 26.52346975981810354118024796404, 27.44228727224615240671256593290, 28.246342737857245819606675402719

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