Properties

Degree 1
Conductor 149
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  + (0.660 + 0.750i)2-s + (0.0424 − 0.999i)3-s + (−0.127 + 0.991i)4-s + (0.942 + 0.333i)5-s + (0.778 − 0.628i)6-s + (0.372 + 0.927i)7-s + (−0.828 + 0.559i)8-s + (−0.996 − 0.0848i)9-s + (0.372 + 0.927i)10-s + (−0.127 − 0.991i)11-s + (0.985 + 0.169i)12-s + (0.524 + 0.851i)13-s + (−0.450 + 0.892i)14-s + (0.372 − 0.927i)15-s + (−0.967 − 0.251i)16-s + (0.0424 − 0.999i)17-s + ⋯
L(s,χ)  = 1  + (0.660 + 0.750i)2-s + (0.0424 − 0.999i)3-s + (−0.127 + 0.991i)4-s + (0.942 + 0.333i)5-s + (0.778 − 0.628i)6-s + (0.372 + 0.927i)7-s + (−0.828 + 0.559i)8-s + (−0.996 − 0.0848i)9-s + (0.372 + 0.927i)10-s + (−0.127 − 0.991i)11-s + (0.985 + 0.169i)12-s + (0.524 + 0.851i)13-s + (−0.450 + 0.892i)14-s + (0.372 − 0.927i)15-s + (−0.967 − 0.251i)16-s + (0.0424 − 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.674 + 0.738i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.674 + 0.738i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(149\)
\( \varepsilon \)  =  $0.674 + 0.738i$
motivic weight  =  \(0\)
character  :  $\chi_{149} (63, \cdot )$
Sato-Tate  :  $\mu(37)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 149,\ (0:\ ),\ 0.674 + 0.738i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(1.564440122 + 0.6898775606i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(1.564440122 + 0.6898775606i\)
\(L(\chi,1)\)  \(\approx\)  \(1.502782800 + 0.4512739053i\)
\(L(1,\chi)\)  \(\approx\)  \(1.502782800 + 0.4512739053i\)

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

−28.153429129495495099590528761676, −27.32894824853669912266944211921, −26.04571640297470225169159449140, −25.137972249212521755588982591991, −23.67040636884435369788540439998, −22.92922514780528629442830213918, −21.83226068987834385289851263940, −21.04594773593280486445079849955, −20.42245036391497155499787460786, −19.578096185083936390681397193702, −17.73631380642843565747641406992, −17.16364739743872404114459583248, −15.55099773175910079296390307674, −14.75876999264957864245225114868, −13.63443473056697088179703607855, −12.89177531715745079180350322367, −11.338243205763557270472507066946, −10.31900541728878310265653551487, −9.86041092490758161099642901172, −8.47214664942219531456386562825, −6.4100511925636393902262410924, −5.13481361345915880532297920847, −4.398824528498176495105027944102, −3.07111801341657507652543021246, −1.52893556428981031573083526456, 2.014197317724882381051038365400, 3.124885700087714917570367019472, 5.175119123311053838445947244739, 6.087612966294437749755186897259, 6.82781774776939691934293933394, 8.30481650542040359779697463754, 9.031337037244259794189552403551, 11.14867725409017810193889218519, 12.14314686726914977541260138200, 13.24098883708434857804550888385, 14.04297528928685580847397208209, 14.72699237631396226376145757549, 16.26174337880038905413952993272, 17.17957880636481609093727615777, 18.45480483072708430193639741059, 18.6470737058751409017634653677, 20.72314646837780590720637951249, 21.5051295570635873958689081308, 22.46575626492094927240575407904, 23.514100591041813717064402965, 24.52954068440255602799397887314, 25.04964698849317849431892467194, 25.861113034177333865875321841584, 26.92722025116359866038363064796, 28.566420498886524736651463480643

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