Properties

Label 2-189-189.38-c1-0-21
Degree $2$
Conductor $189$
Sign $-0.996 + 0.0786i$
Analytic cond. $1.50917$
Root an. cond. $1.22848$
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  + (0.491 + 0.0867i)2-s + (−1.45 − 0.934i)3-s + (−1.64 − 0.598i)4-s + (9.76e−5 + 0.000553i)5-s + (−0.635 − 0.586i)6-s + (−2.32 + 1.26i)7-s + (−1.62 − 0.936i)8-s + (1.25 + 2.72i)9-s + 0.000280i·10-s + (−4.66 − 0.822i)11-s + (1.83 + 2.41i)12-s + (−1.77 − 2.10i)13-s + (−1.25 + 0.421i)14-s + (0.000375 − 0.000898i)15-s + (1.96 + 1.64i)16-s − 0.800·17-s + ⋯
L(s)  = 1  + (0.347 + 0.0613i)2-s + (−0.841 − 0.539i)3-s + (−0.822 − 0.299i)4-s + (4.36e−5 + 0.000247i)5-s + (−0.259 − 0.239i)6-s + (−0.878 + 0.478i)7-s + (−0.573 − 0.331i)8-s + (0.417 + 0.908i)9-s + 8.87e−5i·10-s + (−1.40 − 0.247i)11-s + (0.530 + 0.695i)12-s + (−0.490 − 0.585i)13-s + (−0.334 + 0.112i)14-s + (9.68e−5 − 0.000232i)15-s + (0.491 + 0.412i)16-s − 0.194·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0786i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.996 + 0.0786i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ -0.996 + 0.0786i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00810226 - 0.205817i\)
\(L(\frac12)\) \(\approx\) \(0.00810226 - 0.205817i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.45 + 0.934i)T \)
7 \( 1 + (2.32 - 1.26i)T \)
good2 \( 1 + (-0.491 - 0.0867i)T + (1.87 + 0.684i)T^{2} \)
5 \( 1 + (-9.76e-5 - 0.000553i)T + (-4.69 + 1.71i)T^{2} \)
11 \( 1 + (4.66 + 0.822i)T + (10.3 + 3.76i)T^{2} \)
13 \( 1 + (1.77 + 2.10i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + 0.800T + 17T^{2} \)
19 \( 1 + 5.20iT - 19T^{2} \)
23 \( 1 + (-1.01 - 1.21i)T + (-3.99 + 22.6i)T^{2} \)
29 \( 1 + (5.40 - 6.44i)T + (-5.03 - 28.5i)T^{2} \)
31 \( 1 + (-2.76 + 7.58i)T + (-23.7 - 19.9i)T^{2} \)
37 \( 1 + (1.02 - 1.78i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.20 + 4.36i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (6.39 - 2.32i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (6.96 - 2.53i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (9.11 + 5.26i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (10.1 - 8.49i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-1.24 - 3.41i)T + (-46.7 + 39.2i)T^{2} \)
67 \( 1 + (-0.369 - 2.09i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (-7.56 + 4.37i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.22 + 3.59i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.447 + 2.53i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (1.47 + 1.23i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + 3.23T + 89T^{2} \)
97 \( 1 + (0.614 + 1.68i)T + (-74.3 + 62.3i)T^{2} \)
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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

−12.50603571401326710814074616147, −11.12571567727718635878545945996, −10.20716042661905357653628250988, −9.182920709645092501584475642797, −7.896459206942145630799875176582, −6.62061757200661074804374455740, −5.54076069893551182345969316532, −4.85476251356444731083411704659, −2.88559420094971775398608899728, −0.17659240921627613581416666065, 3.27371837768026646820302351243, 4.47876287123650789005771093627, 5.38089937921213682004077827180, 6.63325034811160704222737314265, 7.968939857215107533388335355529, 9.411716569833568169856651956748, 10.04127473417107384121666901572, 11.03586155525315483177671282663, 12.39871817141594026073549418145, 12.77234538674405968400472500915

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