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Properties

Label 1-187-187.135-r0-0-0
Degree $1$
Conductor $187$
Sign $-0.822 - 0.568i$
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  + (0.309 − 0.951i)2-s + (0.809 − 0.587i)3-s + (−0.809 − 0.587i)4-s + (−0.309 − 0.951i)5-s + (−0.309 − 0.951i)6-s + (0.809 + 0.587i)7-s + (−0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s − 10-s − 12-s + (0.309 − 0.951i)13-s + (0.809 − 0.587i)14-s + (−0.809 − 0.587i)15-s + (0.309 + 0.951i)16-s + (−0.809 − 0.587i)18-s + (−0.809 + 0.587i)19-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)2-s + (0.809 − 0.587i)3-s + (−0.809 − 0.587i)4-s + (−0.309 − 0.951i)5-s + (−0.309 − 0.951i)6-s + (0.809 + 0.587i)7-s + (−0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s − 10-s − 12-s + (0.309 − 0.951i)13-s + (0.809 − 0.587i)14-s + (−0.809 − 0.587i)15-s + (0.309 + 0.951i)16-s + (−0.809 − 0.587i)18-s + (−0.809 + 0.587i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.822 - 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.822 - 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $-0.822 - 0.568i$
Analytic conductor: \(0.868424\)
Root analytic conductor: \(0.868424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{187} (135, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 187,\ (0:\ ),\ -0.822 - 0.568i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4644397369 - 1.489411206i\)
\(L(\frac12)\) \(\approx\) \(0.4644397369 - 1.489411206i\)
\(L(1)\) \(\approx\) \(0.9235892017 - 1.061971360i\)
\(L(1)\) \(\approx\) \(0.9235892017 - 1.061971360i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.309 - 0.951i)T \)
3 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 + (-0.309 - 0.951i)T \)
7 \( 1 + (0.809 + 0.587i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 - T \)
29 \( 1 + (0.809 + 0.587i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (0.809 + 0.587i)T \)
41 \( 1 + (0.809 - 0.587i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.809 + 0.587i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (-0.809 - 0.587i)T \)
61 \( 1 + (-0.309 - 0.951i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (-0.309 + 0.951i)T \)
83 \( 1 + (0.309 + 0.951i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.309 + 0.951i)T \)
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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

−27.211232736020471949730578065079, −26.164425602223371179800876282480, −26.11661052339727095470064771720, −24.72701929830827808647612577090, −23.79180142731131851436851106472, −22.93799895610177399850048359753, −21.73975010082828158742150178434, −21.23250365033328249377134501351, −19.855733638846699719271476283853, −18.78854400457966803837304888887, −17.783271796228843377783109708684, −16.6130666848052117918058383487, −15.65319686755953486857888643998, −14.73592183994211991128394754995, −14.17982930900093277452509866596, −13.34593413168654954677607032178, −11.59448316683873918538015353654, −10.48379689779147446583154013074, −9.251094170247177456604746116354, −8.10030683125103316499596766583, −7.38858592160054633957563327262, −6.1814116494828542774742737087, −4.448253002190426835997412416438, −3.95936017276110303204599983654, −2.43917835311903420765223469035, 1.16275889182179303888099956521, 2.227104775435503382718566402723, 3.58560461966197999905486621440, 4.75680435870357355307297000119, 5.96623762359580047185786909246, 8.05622567153993948660129060201, 8.498803994898393077950846363609, 9.6158454466113148524287847437, 11.01215094075554531402242091352, 12.34460306922114471193343623199, 12.59894470270956030536669005493, 13.85040019816998435074001130463, 14.722033427972478643344030033899, 15.752327411178535059330153000118, 17.56228374508598839648056049610, 18.26984114520173083534730146966, 19.35939758224228461208074699294, 20.1415756760383898689307867520, 20.84877544448852810475285330240, 21.616435045666731379747324235047, 23.11052385840099769644800518341, 23.94557911256631289991328969998, 24.69408222910815761136167950464, 25.67682727523069841716754360466, 27.27778510397674059357870174365

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