Properties

Degree 1
Conductor $ 2^{2} \cdot 19 $
Sign $0.992 - 0.120i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.939 + 0.342i)3-s + (0.173 − 0.984i)5-s + (0.5 + 0.866i)7-s + (0.766 − 0.642i)9-s + (0.5 − 0.866i)11-s + (0.939 + 0.342i)13-s + (0.173 + 0.984i)15-s + (0.766 + 0.642i)17-s + (−0.766 − 0.642i)21-s + (−0.173 − 0.984i)23-s + (−0.939 − 0.342i)25-s + (−0.5 + 0.866i)27-s + (−0.766 + 0.642i)29-s + (−0.5 − 0.866i)31-s + (−0.173 + 0.984i)33-s + ⋯
L(s,χ)  = 1  + (−0.939 + 0.342i)3-s + (0.173 − 0.984i)5-s + (0.5 + 0.866i)7-s + (0.766 − 0.642i)9-s + (0.5 − 0.866i)11-s + (0.939 + 0.342i)13-s + (0.173 + 0.984i)15-s + (0.766 + 0.642i)17-s + (−0.766 − 0.642i)21-s + (−0.173 − 0.984i)23-s + (−0.939 − 0.342i)25-s + (−0.5 + 0.866i)27-s + (−0.766 + 0.642i)29-s + (−0.5 − 0.866i)31-s + (−0.173 + 0.984i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.992 - 0.120i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 76 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.992 - 0.120i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(76\)    =    \(2^{2} \cdot 19\)
\( \varepsilon \)  =  $0.992 - 0.120i$
motivic weight  =  \(0\)
character  :  $\chi_{76} (15, \cdot )$
Sato-Tate  :  $\mu(18)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 76,\ (0:\ ),\ 0.992 - 0.120i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.8135824987 - 0.04903977352i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.8135824987 - 0.04903977352i\)
\(L(\chi,1)\)  \(\approx\)  \(0.8890535618 + 0.01835378084i\)
\(L(1,\chi)\)  \(\approx\)  \(0.8890535618 + 0.01835378084i\)

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

−30.93203897624112577887447555378, −30.08166788066724711132897728453, −29.53707602879429471948455519356, −28.03398395302858755309407811843, −27.28338056096053307003648750105, −25.968040538201862069060932334053, −24.85565164898485520735629335208, −23.2972217558275348145847923915, −23.01099611950523159293441405803, −21.750625251265996760353467417601, −20.45657833054174825351098652330, −18.976383269193511961462989099337, −17.915022662519050517483964441084, −17.24886603199047623073591687519, −15.83534712446400949601791846478, −14.41342742065765652512589367934, −13.33452569963245920101842441130, −11.80809742068884956155103291722, −10.8784468559433808113359796931, −9.88861948418508800898628498257, −7.62785468129583486173104795162, −6.80925897549110393407668749628, −5.430224701022469086446899090009, −3.78841354528954967058726514752, −1.58702375544705554397572365317, 1.39845297491799363830659869704, 3.96460373978699684551707954807, 5.36491213542093143253674175452, 6.17975227323914479848500385497, 8.338402880522388872657769401893, 9.342121317119082885771471483366, 10.946536249349562587317372924884, 11.93221427076666262092298207059, 12.95508117444841774265126700550, 14.60715654847674236264645419017, 16.06512154665685059092719617992, 16.70317037301703992759909234989, 17.917253339658016599352929741488, 19.00817176689478563983534589986, 20.79903745186219519179695047777, 21.40359792139792752758181316974, 22.51610340372847266074200960764, 23.94658566681937993078907446017, 24.4711865432204264672264276805, 25.93230084685762257602500197222, 27.53566993659275931931653835709, 28.012025049164564742272277983332, 28.927597566204550918191750199662, 30.074846011114636174964858939186, 31.49677583617361009120981768050

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