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Properties

Label 2-189-21.5-c1-0-6
Degree $2$
Conductor $189$
Sign $-0.244 + 0.969i$
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  + (−1.58 − 0.917i)2-s + (0.682 + 1.18i)4-s + (1.90 − 3.29i)5-s + (2.11 + 1.58i)7-s + 1.16i·8-s + (−6.04 + 3.49i)10-s + (−1.00 + 0.582i)11-s − 5.54i·13-s + (−1.90 − 4.46i)14-s + (2.43 − 4.21i)16-s + (−1.58 − 2.75i)17-s + (−0.546 − 0.315i)19-s + 5.19·20-s + 2.13·22-s + (1.52 + 0.880i)23-s + ⋯
L(s)  = 1  + (−1.12 − 0.648i)2-s + (0.341 + 0.590i)4-s + (0.851 − 1.47i)5-s + (0.799 + 0.600i)7-s + 0.412i·8-s + (−1.91 + 1.10i)10-s + (−0.304 + 0.175i)11-s − 1.53i·13-s + (−0.508 − 1.19i)14-s + (0.608 − 1.05i)16-s + (−0.385 − 0.667i)17-s + (−0.125 − 0.0724i)19-s + 1.16·20-s + 0.455·22-s + (0.318 + 0.183i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.244 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.244 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.244 + 0.969i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ -0.244 + 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.474412 - 0.608977i\)
\(L(\frac12)\) \(\approx\) \(0.474412 - 0.608977i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-2.11 - 1.58i)T \)
good2 \( 1 + (1.58 + 0.917i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-1.90 + 3.29i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.00 - 0.582i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.54iT - 13T^{2} \)
17 \( 1 + (1.58 + 2.75i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.546 + 0.315i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.52 - 0.880i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.83iT - 29T^{2} \)
31 \( 1 + (3.70 - 2.13i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.11 - 3.66i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 4.56T + 41T^{2} \)
43 \( 1 - 7.23T + 43T^{2} \)
47 \( 1 + (-1.27 + 2.20i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.0627 - 0.0362i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.49 - 6.04i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.00 - 4.04i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.54 + 4.41i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.76iT - 71T^{2} \)
73 \( 1 + (-4.84 + 2.79i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.54 - 14.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.1T + 83T^{2} \)
89 \( 1 + (5.77 - 10.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 0.688iT - 97T^{2} \)
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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

−12.24100572266102278514617262732, −11.07960030771103631633644937770, −10.17837244598068526567043894519, −9.166448847613392526791079482069, −8.649903331779284265810873682997, −7.72560382847995855910274643516, −5.52598764826615233312886250904, −5.02521387992017137050014598221, −2.44542988429271814643989586312, −1.10171270149085016099708516194, 2.03115188513940804053021374083, 4.02667237737883887843363490922, 6.02409314810906355043850349659, 6.89156237378885966902381249231, 7.62462041037946209355932462961, 8.857297711851824670719897245807, 9.817533785282490120865250661664, 10.68160151428142229955619442695, 11.32963558550462242488845042965, 13.07801224736333968868093401159

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