Properties

Degree 1
Conductor $ 11 \cdot 17 $
Sign $0.240 - 0.970i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.987 − 0.156i)2-s + (0.649 − 0.760i)3-s + (0.951 − 0.309i)4-s + (−0.233 − 0.972i)5-s + (0.522 − 0.852i)6-s + (−0.760 + 0.649i)7-s + (0.891 − 0.453i)8-s + (−0.156 − 0.987i)9-s + (−0.382 − 0.923i)10-s + (0.382 − 0.923i)12-s + (−0.587 + 0.809i)13-s + (−0.649 + 0.760i)14-s + (−0.891 − 0.453i)15-s + (0.809 − 0.587i)16-s + (−0.309 − 0.951i)18-s + (0.453 + 0.891i)19-s + ⋯
L(s,χ)  = 1  + (0.987 − 0.156i)2-s + (0.649 − 0.760i)3-s + (0.951 − 0.309i)4-s + (−0.233 − 0.972i)5-s + (0.522 − 0.852i)6-s + (−0.760 + 0.649i)7-s + (0.891 − 0.453i)8-s + (−0.156 − 0.987i)9-s + (−0.382 − 0.923i)10-s + (0.382 − 0.923i)12-s + (−0.587 + 0.809i)13-s + (−0.649 + 0.760i)14-s + (−0.891 − 0.453i)15-s + (0.809 − 0.587i)16-s + (−0.309 − 0.951i)18-s + (0.453 + 0.891i)19-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.240 - 0.970i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.240 - 0.970i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(187\)    =    \(11 \cdot 17\)
\( \varepsilon \)  =  $0.240 - 0.970i$
motivic weight  =  \(0\)
character  :  $\chi_{187} (156, \cdot )$
Sato-Tate  :  $\mu(80)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 187,\ (0:\ ),\ 0.240 - 0.970i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.783929918 - 1.395993384i$
$L(\frac12,\chi)$  $\approx$  $1.783929918 - 1.395993384i$
$L(\chi,1)$  $\approx$  1.777278511 - 0.8398642865i
$L(1,\chi)$  $\approx$  1.777278511 - 0.8398642865i

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

−26.9690538334312972270070265280, −26.29615074722616749267830568426, −25.57435660638711992512923083335, −24.57613917208973336732954300588, −23.228015475344955419296015386782, −22.485544013925941261817742781671, −21.92305361729970295946583134418, −20.75044400444620888459436702412, −19.835349440146110158720164019511, −19.200968819357438824538125634472, −17.39533554429380076823527816811, −16.26365837911960740678433483625, −15.36588549644027215074989368950, −14.748304737665381573024444576586, −13.72099954082421805012912567065, −12.929919210971771167057909628388, −11.3290434641772528491682313345, −10.55373805138606852660824043807, −9.527774312507293456392378247117, −7.73016160955459887593548375065, −7.05157097976777511929514429410, −5.62872919124946561298569979217, −4.29998334803704342828428335057, −3.29592617144205937011838785906, −2.60564449726394657147645008015, 1.47551278660293434918996389437, 2.7173215722787818482222739627, 3.88582528603090958905076498005, 5.246960540721202811527106884727, 6.40662238020713754119087360111, 7.4719428523505563117079400634, 8.77017701181596396353827258309, 9.775645095524249384551124648882, 11.66919094948053677677130603363, 12.374393562197052032148087274928, 13.015920216693511546632833957375, 14.02251319569314831045256035151, 15.03761927095922808752748040234, 16.035918652038443484238273396428, 17.01329057709522350192140968106, 18.80645827044326327163088019315, 19.35017383382108644120315961039, 20.3720572695940156778933004963, 21.06852526322060376788632219282, 22.2571041730158296176234698619, 23.31827798762159340736284088270, 24.16748799137731563410823595859, 24.89660569768987912046834252411, 25.502430176302917250525223845202, 26.82389368738954299267166743566

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