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Properties

Label 1-187-187.63-r0-0-0
Degree $1$
Conductor $187$
Sign $0.160 + 0.987i$
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  + (−0.453 − 0.891i)2-s + (0.233 + 0.972i)3-s + (−0.587 + 0.809i)4-s + (−0.0784 + 0.996i)5-s + (0.760 − 0.649i)6-s + (0.972 + 0.233i)7-s + (0.987 + 0.156i)8-s + (−0.891 + 0.453i)9-s + (0.923 − 0.382i)10-s + (−0.923 − 0.382i)12-s + (−0.951 − 0.309i)13-s + (−0.233 − 0.972i)14-s + (−0.987 + 0.156i)15-s + (−0.309 − 0.951i)16-s + (0.809 + 0.587i)18-s + (−0.156 + 0.987i)19-s + ⋯
L(s)  = 1  + (−0.453 − 0.891i)2-s + (0.233 + 0.972i)3-s + (−0.587 + 0.809i)4-s + (−0.0784 + 0.996i)5-s + (0.760 − 0.649i)6-s + (0.972 + 0.233i)7-s + (0.987 + 0.156i)8-s + (−0.891 + 0.453i)9-s + (0.923 − 0.382i)10-s + (−0.923 − 0.382i)12-s + (−0.951 − 0.309i)13-s + (−0.233 − 0.972i)14-s + (−0.987 + 0.156i)15-s + (−0.309 − 0.951i)16-s + (0.809 + 0.587i)18-s + (−0.156 + 0.987i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.160 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.160 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $0.160 + 0.987i$
Analytic conductor: \(0.868424\)
Root analytic conductor: \(0.868424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{187} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 187,\ (0:\ ),\ 0.160 + 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6549149893 + 0.5570162508i\)
\(L(\frac12)\) \(\approx\) \(0.6549149893 + 0.5570162508i\)
\(L(1)\) \(\approx\) \(0.8247269160 + 0.2225070858i\)
\(L(1)\) \(\approx\) \(0.8247269160 + 0.2225070858i\)

Euler product

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

−26.83146587901213926955297585215, −25.879584882882073291200373635504, −24.733978978959336243424930434, −24.2677606755849591718091302594, −23.7719041776909131110287090597, −22.56390593589146694052260984950, −20.974553320848276715552117219988, −19.924367027010634451265790452293, −19.18909557583171862267114790362, −17.98185636460533043609072660712, −17.30635820868169870637906166549, −16.55268660376848371697967838420, −15.121379203456830774172067524796, −14.2747540694001941660949954667, −13.32388087093027497075105783327, −12.293096997551654891658967259682, −11.0311052839084262839825689608, −9.37599333383892329890729367545, −8.53896687964862687103919254246, −7.703247429952338119253599758461, −6.79175551125867629237834632621, −5.374361694088164854526961013831, −4.487486181865279519809383169443, −2.06313127767612693240244618272, −0.78982438381286183428158310339, 2.06382753242180240759473634499, 3.11955035581829184174874532100, 4.22681401454747843420746326947, 5.43317478344575355637999589459, 7.481085722640077158665979382657, 8.35956254236424507048497481945, 9.633694101367345752190913038119, 10.40174568546002681807373466823, 11.25610903537734310954481805880, 12.09341127069796154209022666979, 13.81032231588627514880604444501, 14.61352282997732135118671654234, 15.5437466052138755357293126225, 17.03851181335674773247083146396, 17.714211962693405920090126300130, 18.898970366790730183321211494967, 19.68989105908268698365130878945, 20.86664807540715832394194531379, 21.42342497578368970265576526719, 22.33771763991322562706845903655, 23.0716548717323421319785096871, 24.84403091811517558628073977409, 25.89084608984815125537113947443, 26.80017237179959126286484318026, 27.30109095826322651370759070908

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