Properties

Degree 1
Conductor 151
Sign $-0.415 + 0.909i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.669 + 0.743i)2-s + (0.968 + 0.248i)3-s + (−0.104 + 0.994i)4-s + (−0.855 + 0.518i)5-s + (0.463 + 0.886i)6-s + (−0.756 − 0.653i)7-s + (−0.809 + 0.587i)8-s + (0.876 + 0.481i)9-s + (−0.957 − 0.289i)10-s + (−0.268 + 0.963i)11-s + (−0.348 + 0.937i)12-s + (0.783 + 0.621i)13-s + (−0.0209 − 0.999i)14-s + (−0.957 + 0.289i)15-s + (−0.978 − 0.207i)16-s + (0.944 + 0.328i)17-s + ⋯
L(s,χ)  = 1  + (0.669 + 0.743i)2-s + (0.968 + 0.248i)3-s + (−0.104 + 0.994i)4-s + (−0.855 + 0.518i)5-s + (0.463 + 0.886i)6-s + (−0.756 − 0.653i)7-s + (−0.809 + 0.587i)8-s + (0.876 + 0.481i)9-s + (−0.957 − 0.289i)10-s + (−0.268 + 0.963i)11-s + (−0.348 + 0.937i)12-s + (0.783 + 0.621i)13-s + (−0.0209 − 0.999i)14-s + (−0.957 + 0.289i)15-s + (−0.978 − 0.207i)16-s + (0.944 + 0.328i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.415 + 0.909i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.415 + 0.909i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(151\)
\( \varepsilon \)  =  $-0.415 + 0.909i$
motivic weight  =  \(0\)
character  :  $\chi_{151} (47, \cdot )$
Sato-Tate  :  $\mu(75)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 151,\ (0:\ ),\ -0.415 + 0.909i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.9068857803 + 1.411573620i$
$L(\frac12,\chi)$  $\approx$  $0.9068857803 + 1.411573620i$
$L(\chi,1)$  $\approx$  1.229183774 + 0.9620937477i
$L(1,\chi)$  $\approx$  1.229183774 + 0.9620937477i

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

−27.747064825672072170370023848142, −27.06695373845937261947289731667, −25.5356325142389460023457533586, −24.78390777933238973298831828332, −23.65380484513160028864503292476, −22.98369004247506696810093770227, −21.53553660188532690477483737452, −20.83551962498309151102483349509, −19.837899182035809397340955777801, −19.03247245099324180840215878149, −18.52328394618785112661071953629, −16.23476028026283953285648968278, −15.473920926209527680479334364898, −14.4962683869559282286359640204, −13.20561601230763778337893656406, −12.69801113090871638091037142663, −11.60286303141577591760064085925, −10.24495498304996481331496735462, −8.97633057862039544038602656934, −8.17657507846310875471955882278, −6.47195730882199940014944102005, −5.11506281754459391009469114635, −3.47997232092356230723039992875, −3.08934910846040420103923480590, −1.20277434799558850958564721180, 2.66642879735267794832654523014, 3.78937698897102550395530467657, 4.46741747347173207516622312548, 6.50039068998503803564635479459, 7.35484255263644748373341784777, 8.25156982129878285718800096561, 9.56759449543155841870997760016, 10.92290036303000691079999292043, 12.47546464621644823903343874540, 13.336059384661801555691981571738, 14.404386270935941253300649359146, 15.23019424918544597347887854646, 15.95605914783265464881870210406, 17.00656031009008108401297088568, 18.61148200742925879838278410667, 19.50916381833465026847501750912, 20.62382608952211514004363864453, 21.46645930558415065319335005171, 22.94480969304744796305369427508, 23.20663774339076207509214843819, 24.468740507976134513970200389305, 25.85313605553523605616684717757, 25.97739587783901564835743259794, 26.945091638326249988478614808057, 28.120732609008625312651388996257

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