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Properties

Label 2-189-189.101-c1-0-18
Degree $2$
Conductor $189$
Sign $-0.298 + 0.954i$
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  + (1.75 − 2.08i)2-s + (−1.10 + 1.33i)3-s + (−0.940 − 5.33i)4-s + (2.08 − 1.74i)5-s + (0.856 + 4.63i)6-s + (−0.122 + 2.64i)7-s + (−8.05 − 4.65i)8-s + (−0.568 − 2.94i)9-s − 7.40i·10-s + (0.0911 − 0.108i)11-s + (8.16 + 4.62i)12-s + (−0.695 + 1.91i)13-s + (5.29 + 4.88i)14-s + (0.0376 + 4.71i)15-s + (−13.6 + 4.96i)16-s + 4.00·17-s + ⋯
L(s)  = 1  + (1.23 − 1.47i)2-s + (−0.636 + 0.771i)3-s + (−0.470 − 2.66i)4-s + (0.931 − 0.781i)5-s + (0.349 + 1.89i)6-s + (−0.0463 + 0.998i)7-s + (−2.84 − 1.64i)8-s + (−0.189 − 0.981i)9-s − 2.34i·10-s + (0.0274 − 0.0327i)11-s + (2.35 + 1.33i)12-s + (−0.192 + 0.530i)13-s + (1.41 + 1.30i)14-s + (0.00971 + 1.21i)15-s + (−3.40 + 1.24i)16-s + 0.970·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.298 + 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.298 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.298 + 0.954i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ -0.298 + 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10864 - 1.50800i\)
\(L(\frac12)\) \(\approx\) \(1.10864 - 1.50800i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.10 - 1.33i)T \)
7 \( 1 + (0.122 - 2.64i)T \)
good2 \( 1 + (-1.75 + 2.08i)T + (-0.347 - 1.96i)T^{2} \)
5 \( 1 + (-2.08 + 1.74i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (-0.0911 + 0.108i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (0.695 - 1.91i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 - 4.00T + 17T^{2} \)
19 \( 1 - 3.04iT - 19T^{2} \)
23 \( 1 + (0.449 - 1.23i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (-2.12 - 5.84i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (4.18 - 0.738i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-4.69 + 8.12i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.303 - 0.110i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-0.643 + 3.65i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (1.15 - 6.57i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (6.59 + 3.81i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.06 + 2.20i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (11.1 + 1.96i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (1.37 - 1.15i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-6.03 + 3.48i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.67 + 2.12i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (11.4 + 9.62i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (0.966 - 0.351i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + 5.17T + 89T^{2} \)
97 \( 1 + (7.25 + 1.27i)T + (91.1 + 33.1i)T^{2} \)
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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

−12.42043203818800932923441099670, −11.48895468415244982903365169901, −10.53096926857948685108813099883, −9.544980974801195547255002714015, −9.133907656486211128037052225426, −6.06456160636455591706472921716, −5.50725270754809051139528324921, −4.67594730656399651065583312204, −3.29857073796211077914382443059, −1.67082978424396752150744859438, 2.92935547350536007070193442701, 4.61504876128693274895956745291, 5.77877425320820477227379326674, 6.47504076346463991882204836245, 7.29971313239781833650008855071, 8.066554144042852172113110129943, 9.957650225950072635861691293310, 11.21244036401565638948378504787, 12.38002695977875560154535424468, 13.24490384929582558676693654141

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