Properties

Label 2-187-17.4-c1-0-4
Degree $2$
Conductor $187$
Sign $-0.331 - 0.943i$
Analytic cond. $1.49320$
Root an. cond. $1.22196$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.48i·2-s + (−0.527 + 0.527i)3-s − 0.209·4-s + (0.786 − 0.786i)5-s + (−0.783 − 0.783i)6-s + (0.754 + 0.754i)7-s + 2.66i·8-s + 2.44i·9-s + (1.16 + 1.16i)10-s + (0.707 + 0.707i)11-s + (0.110 − 0.110i)12-s − 1.70·13-s + (−1.12 + 1.12i)14-s + 0.828i·15-s − 4.37·16-s + (−1.83 − 3.69i)17-s + ⋯
L(s)  = 1  + 1.05i·2-s + (−0.304 + 0.304i)3-s − 0.104·4-s + (0.351 − 0.351i)5-s + (−0.319 − 0.319i)6-s + (0.284 + 0.284i)7-s + 0.940i·8-s + 0.814i·9-s + (0.369 + 0.369i)10-s + (0.213 + 0.213i)11-s + (0.0319 − 0.0319i)12-s − 0.472·13-s + (−0.299 + 0.299i)14-s + 0.214i·15-s − 1.09·16-s + (−0.444 − 0.895i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.331 - 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $-0.331 - 0.943i$
Analytic conductor: \(1.49320\)
Root analytic conductor: \(1.22196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{187} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 187,\ (\ :1/2),\ -0.331 - 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.711774 + 1.00485i\)
\(L(\frac12)\) \(\approx\) \(0.711774 + 1.00485i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.707 - 0.707i)T \)
17 \( 1 + (1.83 + 3.69i)T \)
good2 \( 1 - 1.48iT - 2T^{2} \)
3 \( 1 + (0.527 - 0.527i)T - 3iT^{2} \)
5 \( 1 + (-0.786 + 0.786i)T - 5iT^{2} \)
7 \( 1 + (-0.754 - 0.754i)T + 7iT^{2} \)
13 \( 1 + 1.70T + 13T^{2} \)
19 \( 1 + 7.65iT - 19T^{2} \)
23 \( 1 + (-5.73 - 5.73i)T + 23iT^{2} \)
29 \( 1 + (-0.352 + 0.352i)T - 29iT^{2} \)
31 \( 1 + (-6.88 + 6.88i)T - 31iT^{2} \)
37 \( 1 + (2.07 - 2.07i)T - 37iT^{2} \)
41 \( 1 + (-3.10 - 3.10i)T + 41iT^{2} \)
43 \( 1 + 12.7iT - 43T^{2} \)
47 \( 1 + 7.40T + 47T^{2} \)
53 \( 1 + 4.02iT - 53T^{2} \)
59 \( 1 - 2.32iT - 59T^{2} \)
61 \( 1 + (1.68 + 1.68i)T + 61iT^{2} \)
67 \( 1 - 10.3T + 67T^{2} \)
71 \( 1 + (-6.62 + 6.62i)T - 71iT^{2} \)
73 \( 1 + (2.08 - 2.08i)T - 73iT^{2} \)
79 \( 1 + (9.70 + 9.70i)T + 79iT^{2} \)
83 \( 1 - 11.9iT - 83T^{2} \)
89 \( 1 + 0.543T + 89T^{2} \)
97 \( 1 + (10.0 - 10.0i)T - 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.25979861736563117201320191766, −11.59711971001578563035588453738, −11.17605697624021283931937040215, −9.676997034549089089486522374073, −8.768032696156131352201922370601, −7.56598799106500826630578285068, −6.74323538380585475182756357479, −5.27290407166178191943386955791, −4.90380866049691335247199457068, −2.39915718409127306259557025277, 1.36373669117259041941212931878, 2.94934408834039148748825431713, 4.27422045112317304824318777347, 6.15780419943381494483703962921, 6.85156975323274468964596111333, 8.373934295496365625508893991969, 9.720683720874337568534218927085, 10.49752117207723934827527101861, 11.27439318438696160356680149148, 12.40365220137631395138556278869

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