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Properties

Label 1-187-187.16-r0-0-0
Degree $1$
Conductor $187$
Sign $0.0457 + 0.998i$
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  + (−0.809 + 0.587i)2-s + (−0.309 − 0.951i)3-s + (0.309 − 0.951i)4-s + (0.809 + 0.587i)5-s + (0.809 + 0.587i)6-s + (−0.309 + 0.951i)7-s + (0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s − 10-s − 12-s + (−0.809 + 0.587i)13-s + (−0.309 − 0.951i)14-s + (0.309 − 0.951i)15-s + (−0.809 − 0.587i)16-s + (0.309 − 0.951i)18-s + (0.309 + 0.951i)19-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)2-s + (−0.309 − 0.951i)3-s + (0.309 − 0.951i)4-s + (0.809 + 0.587i)5-s + (0.809 + 0.587i)6-s + (−0.309 + 0.951i)7-s + (0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s − 10-s − 12-s + (−0.809 + 0.587i)13-s + (−0.309 − 0.951i)14-s + (0.309 − 0.951i)15-s + (−0.809 − 0.587i)16-s + (0.309 − 0.951i)18-s + (0.309 + 0.951i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0457 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0457 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $0.0457 + 0.998i$
Analytic conductor: \(0.868424\)
Root analytic conductor: \(0.868424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{187} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 187,\ (0:\ ),\ 0.0457 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4491125050 + 0.4290175490i\)
\(L(\frac12)\) \(\approx\) \(0.4491125050 + 0.4290175490i\)
\(L(1)\) \(\approx\) \(0.6325809975 + 0.1987415146i\)
\(L(1)\) \(\approx\) \(0.6325809975 + 0.1987415146i\)

Euler product

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

−26.88344744134137247341506246665, −26.34694137788202367779598506251, −25.39775912690465607297417773031, −24.24838466874153363806727301955, −22.74386874341848855964868597046, −21.91185186824383924099081977953, −21.06355255224887953754739695052, −20.21248265112554502913770632397, −19.60894588824483727245627424104, −17.812247777054225520323067125378, −17.33961617980360074463321945409, −16.51310673542761519552130859633, −15.63793948935621971532196066730, −14.03327445474222278242975671181, −12.940523194102971660897325644907, −11.82187754432679011958275761340, −10.600640518631641662970539960461, −9.926005615837965841158644174702, −9.2394121097147367954589544947, −7.94685697407706865275178410249, −6.50471146826718539149526431041, −5.02013703405967672600126337700, −3.8719837892065451983335723681, −2.50091102733110429398830595294, −0.652993461999274289293731725537, 1.695236770454120627073157657, 2.582101109883165187851546286213, 5.342271608540210556794444145317, 6.119051427375923167652865909537, 6.946612093651667822146300072533, 8.056578894368905669213717660416, 9.26321964011571838903404875031, 10.19148538042744842532823019151, 11.47719823187278921504473910650, 12.47906168171702849561902201297, 13.938601468160339271492392099080, 14.566567417335837828339279318767, 15.90322425669586204750874918778, 17.034338447902551688682870194744, 17.75536711682748901638559666957, 18.78291611046730346575872210718, 18.99175172532589397781044223483, 20.42748770419177454661763632282, 21.97543973088385777798954584865, 22.6858976907953684739347469234, 24.0119797409530183339282937897, 24.66568244868988362441658623459, 25.525034191600289216370147715683, 26.10666956577327852359629051624, 27.407698028563366328216197904576

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