Properties

Label 2-189-189.4-c1-0-11
Degree $2$
Conductor $189$
Sign $0.927 + 0.373i$
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  + (−0.134 − 0.764i)2-s + (1.54 + 0.776i)3-s + (1.31 − 0.477i)4-s + (1.10 − 0.402i)5-s + (0.385 − 1.28i)6-s + (−0.975 + 2.45i)7-s + (−1.31 − 2.28i)8-s + (1.79 + 2.40i)9-s + (−0.456 − 0.791i)10-s + (−3.93 − 1.43i)11-s + (2.40 + 0.280i)12-s + (−3.70 + 1.34i)13-s + (2.01 + 0.414i)14-s + (2.02 + 0.236i)15-s + (0.570 − 0.478i)16-s + (0.400 + 0.692i)17-s + ⋯
L(s)  = 1  + (−0.0953 − 0.540i)2-s + (0.893 + 0.448i)3-s + (0.656 − 0.238i)4-s + (0.494 − 0.179i)5-s + (0.157 − 0.526i)6-s + (−0.368 + 0.929i)7-s + (−0.466 − 0.807i)8-s + (0.597 + 0.801i)9-s + (−0.144 − 0.250i)10-s + (−1.18 − 0.431i)11-s + (0.693 + 0.0809i)12-s + (−1.02 + 0.374i)13-s + (0.537 + 0.110i)14-s + (0.522 + 0.0609i)15-s + (0.142 − 0.119i)16-s + (0.0970 + 0.168i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.373i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 + 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.927 + 0.373i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.927 + 0.373i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60100 - 0.309778i\)
\(L(\frac12)\) \(\approx\) \(1.60100 - 0.309778i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.54 - 0.776i)T \)
7 \( 1 + (0.975 - 2.45i)T \)
good2 \( 1 + (0.134 + 0.764i)T + (-1.87 + 0.684i)T^{2} \)
5 \( 1 + (-1.10 + 0.402i)T + (3.83 - 3.21i)T^{2} \)
11 \( 1 + (3.93 + 1.43i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (3.70 - 1.34i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-0.400 - 0.692i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.30 + 2.26i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.28 + 7.30i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (2.12 + 0.773i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-7.31 + 2.66i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + 2.78T + 37T^{2} \)
41 \( 1 + (2.90 - 1.05i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-1.78 - 10.1i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-0.726 - 0.264i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (5.10 - 8.84i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7.53 - 6.32i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-7.00 - 2.54i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-1.80 + 10.2i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (0.981 - 1.69i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 5.19T + 73T^{2} \)
79 \( 1 + (1.05 + 6.00i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-3.56 - 1.29i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (-4.65 + 8.07i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.984 + 5.58i)T + (-91.1 + 33.1i)T^{2} \)
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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

−12.56720233117793032086564058571, −11.45004603672385903381490764802, −10.29594386661003433679415699850, −9.709880971238659850681449940707, −8.757289070790788407142529474031, −7.53848372428912642679039242799, −6.14010612293671376664489931326, −4.88398316207139591287025532686, −2.94303101682702403054246704182, −2.28748636014933879445707554359, 2.20093815330684137820102621422, 3.38606206559191881915458894358, 5.37472646791510386610882294401, 6.81616048100114965810648653013, 7.45539349373919116295511577713, 8.179732639663132664922614559319, 9.744440315470707622862452788067, 10.32626684793932602241145808021, 11.84345654215534454193534411853, 12.82477829719142123651869886977

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