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Properties

Label 1-187-187.6-r0-0-0
Degree $1$
Conductor $187$
Sign $0.240 + 0.970i$
Analytic cond. $0.868424$
Root an. cond. $0.868424$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $0.240 + 0.970i$
Analytic conductor: \(0.868424\)
Root analytic conductor: \(0.868424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{187} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 187,\ (0:\ ),\ 0.240 + 0.970i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.783929918 + 1.395993384i\)
\(L(\frac12)\) \(\approx\) \(1.783929918 + 1.395993384i\)
\(L(1)\) \(\approx\) \(1.777278511 + 0.8398642865i\)
\(L(1)\) \(\approx\) \(1.777278511 + 0.8398642865i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.987 + 0.156i)T \)
3 \( 1 + (0.649 + 0.760i)T \)
5 \( 1 + (-0.233 + 0.972i)T \)
7 \( 1 + (-0.760 - 0.649i)T \)
13 \( 1 + (-0.587 - 0.809i)T \)
19 \( 1 + (0.453 - 0.891i)T \)
23 \( 1 + (0.923 + 0.382i)T \)
29 \( 1 + (-0.996 + 0.0784i)T \)
31 \( 1 + (0.522 - 0.852i)T \)
37 \( 1 + (-0.0784 - 0.996i)T \)
41 \( 1 + (0.996 + 0.0784i)T \)
43 \( 1 + (-0.707 + 0.707i)T \)
47 \( 1 + (0.951 - 0.309i)T \)
53 \( 1 + (-0.987 - 0.156i)T \)
59 \( 1 + (0.453 + 0.891i)T \)
61 \( 1 + (0.852 - 0.522i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.972 - 0.233i)T \)
73 \( 1 + (0.996 - 0.0784i)T \)
79 \( 1 + (-0.972 + 0.233i)T \)
83 \( 1 + (-0.156 - 0.987i)T \)
89 \( 1 + iT \)
97 \( 1 + (-0.852 - 0.522i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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