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Properties

Label 1-187-187.41-r0-0-0
Degree $1$
Conductor $187$
Sign $0.521 - 0.853i$
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  + (0.891 − 0.453i)2-s + (0.852 + 0.522i)3-s + (0.587 − 0.809i)4-s + (−0.649 − 0.760i)5-s + (0.996 + 0.0784i)6-s + (−0.522 − 0.852i)7-s + (0.156 − 0.987i)8-s + (0.453 + 0.891i)9-s + (−0.923 − 0.382i)10-s + (0.923 − 0.382i)12-s + (0.951 + 0.309i)13-s + (−0.852 − 0.522i)14-s + (−0.156 − 0.987i)15-s + (−0.309 − 0.951i)16-s + (0.809 + 0.587i)18-s + (−0.987 − 0.156i)19-s + ⋯
L(s)  = 1  + (0.891 − 0.453i)2-s + (0.852 + 0.522i)3-s + (0.587 − 0.809i)4-s + (−0.649 − 0.760i)5-s + (0.996 + 0.0784i)6-s + (−0.522 − 0.852i)7-s + (0.156 − 0.987i)8-s + (0.453 + 0.891i)9-s + (−0.923 − 0.382i)10-s + (0.923 − 0.382i)12-s + (0.951 + 0.309i)13-s + (−0.852 − 0.522i)14-s + (−0.156 − 0.987i)15-s + (−0.309 − 0.951i)16-s + (0.809 + 0.587i)18-s + (−0.987 − 0.156i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.521 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.521 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $0.521 - 0.853i$
Analytic conductor: \(0.868424\)
Root analytic conductor: \(0.868424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{187} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 187,\ (0:\ ),\ 0.521 - 0.853i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.891842734 - 1.061406952i\)
\(L(\frac12)\) \(\approx\) \(1.891842734 - 1.061406952i\)
\(L(1)\) \(\approx\) \(1.786450744 - 0.6192347491i\)
\(L(1)\) \(\approx\) \(1.786450744 - 0.6192347491i\)

Euler product

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

−26.91566577372191018025777730444, −25.997480484845952087610467782651, −25.398643492944073032151408328706, −24.542801163777228191073074330037, −23.430416589061112485717556325221, −22.79898453831403438267698014676, −21.67824014751126576722916312881, −20.743139367574208128712594392590, −19.607476282273643142606632347797, −18.78704378940363956563773258427, −17.79932202388671501572614326600, −16.07815225792444376708650917781, −15.39791841160091198013611113688, −14.631469060793247308428572387120, −13.671655770562470893362574201041, −12.6172751553006521281988813499, −11.898951435218481875400549223893, −10.5099105796735087606131652895, −8.6966698087664493025558182779, −8.02149141460003022081335027545, −6.72181993479585127829647875730, −6.10989173622427237999273731271, −4.21010846649833467818038667892, −3.18620219655779440404020548400, −2.33305038689100124463708966456, 1.43166656939933208222960552110, 3.14391041444516446485606313013, 4.02103048786956599166843423562, 4.75383431063088752161135262088, 6.41604281606600474638934527385, 7.77474225839543917903166666256, 8.98590091856343017107539334763, 10.1349252967481801530464068419, 11.07472094260913836947876182901, 12.35678265836245304443910984128, 13.4183561726247271084426719236, 13.93717712494240294413090868304, 15.37737530484963946644100645849, 15.86968904605276157231280896245, 16.90880881749397556791136866334, 19.008457693852017488432554421496, 19.59558511411519501569413328794, 20.458556034017130356327216814704, 21.03488679737804321230808112056, 22.12338192364603961271812962250, 23.354417533061928506832008138842, 23.82465449494566277071661601996, 25.11424031310096260276703902828, 25.910470732499797831718794533503, 27.180123193082296617892334216501

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