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Properties

Label 1-187-187.10-r0-0-0
Degree $1$
Conductor $187$
Sign $-0.518 + 0.855i$
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  + (0.707 + 0.707i)2-s + (0.382 + 0.923i)3-s + i·4-s + (0.923 − 0.382i)5-s + (−0.382 + 0.923i)6-s + (−0.923 − 0.382i)7-s + (−0.707 + 0.707i)8-s + (−0.707 + 0.707i)9-s + (0.923 + 0.382i)10-s + (−0.923 + 0.382i)12-s + i·13-s + (−0.382 − 0.923i)14-s + (0.707 + 0.707i)15-s − 16-s − 18-s + (0.707 + 0.707i)19-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + (0.382 + 0.923i)3-s + i·4-s + (0.923 − 0.382i)5-s + (−0.382 + 0.923i)6-s + (−0.923 − 0.382i)7-s + (−0.707 + 0.707i)8-s + (−0.707 + 0.707i)9-s + (0.923 + 0.382i)10-s + (−0.923 + 0.382i)12-s + i·13-s + (−0.382 − 0.923i)14-s + (0.707 + 0.707i)15-s − 16-s − 18-s + (0.707 + 0.707i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.518 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.518 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $-0.518 + 0.855i$
Analytic conductor: \(0.868424\)
Root analytic conductor: \(0.868424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{187} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 187,\ (0:\ ),\ -0.518 + 0.855i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9116742397 + 1.618639054i\)
\(L(\frac12)\) \(\approx\) \(0.9116742397 + 1.618639054i\)
\(L(1)\) \(\approx\) \(1.237265934 + 1.086645215i\)
\(L(1)\) \(\approx\) \(1.237265934 + 1.086645215i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 + (0.382 + 0.923i)T \)
5 \( 1 + (0.923 - 0.382i)T \)
7 \( 1 + (-0.923 - 0.382i)T \)
13 \( 1 + iT \)
19 \( 1 + (0.707 + 0.707i)T \)
23 \( 1 + (0.382 - 0.923i)T \)
29 \( 1 + (0.923 - 0.382i)T \)
31 \( 1 + (-0.382 - 0.923i)T \)
37 \( 1 + (0.382 + 0.923i)T \)
41 \( 1 + (-0.923 - 0.382i)T \)
43 \( 1 + (0.707 - 0.707i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.707 - 0.707i)T \)
59 \( 1 + (0.707 - 0.707i)T \)
61 \( 1 + (0.923 + 0.382i)T \)
67 \( 1 - T \)
71 \( 1 + (0.382 + 0.923i)T \)
73 \( 1 + (-0.923 + 0.382i)T \)
79 \( 1 + (0.382 - 0.923i)T \)
83 \( 1 + (-0.707 - 0.707i)T \)
89 \( 1 + iT \)
97 \( 1 + (-0.923 + 0.382i)T \)
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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

−26.781421951974281897860339869101, −25.37641919920066682554582090536, −25.14823082472824083499358661247, −23.887869257408803134444890773980, −22.87418011080279796378298934118, −22.14324969420103842254560681623, −21.16085824092518593051502583909, −19.9814877320794670755923327235, −19.37007363120193749623696587217, −18.27573730765113105958216480832, −17.64945840103909534493200776646, −15.80064416862879254539007017468, −14.72007383224034872493683971561, −13.74946099620760230254754107358, −13.05570510425338498439834194890, −12.332007800412413373864153462145, −11.030083215269605241412813978771, −9.82908677128068380940435227607, −8.99452432770757928864144336831, −7.170760034104506897691025885067, −6.17838993704674163671615963554, −5.34223954932850529441371987999, −3.204805707129806695732236450789, −2.6973396966872619778855993898, −1.26183389919981151686769278495, 2.45884132340164159146080550191, 3.696017558165778253450350227923, 4.72040071465402479068884613138, 5.83066209893061735520820772545, 6.84865938771963906892589671963, 8.39958547453295643539528185916, 9.364693409212931113598836889973, 10.237386916127130802369647170992, 11.82099411013249014551522887285, 13.11991754464334853454211857688, 13.86292590952817289960427976057, 14.64643783373163173811204854007, 15.93936247424953168554666740722, 16.55487694796758075249998750602, 17.23726802443854487029863566588, 18.806667373175892598810496889057, 20.33310300524521482415684292058, 20.89814536336811372595469027474, 21.96288006315115421848473791503, 22.47473347978849246393486630050, 23.6672393310433654705662255193, 24.85682520939435324169108993867, 25.57410529028977802597566377864, 26.30942326886092559012020327600, 27.05426214206930096674088189174

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