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Properties

Label 2-189-63.38-c1-0-5
Degree 22
Conductor 189189
Sign −0.749+0.662i-0.749 + 0.662i
Analytic cond. 1.509171.50917
Root an. cond. 1.228481.22848
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.09i·2-s − 2.39·4-s + (1.04 − 1.80i)5-s + (2.60 − 0.486i)7-s + 0.819i·8-s + (−3.79 − 2.18i)10-s + (−2.79 + 1.61i)11-s + (−2.68 + 1.55i)13-s + (−1.01 − 5.44i)14-s − 3.06·16-s + (−0.816 + 1.41i)17-s + (4.79 − 2.76i)19-s + (−2.49 + 4.32i)20-s + (3.38 + 5.85i)22-s + (1.00 + 0.580i)23-s + ⋯
L(s)  = 1  − 1.48i·2-s − 1.19·4-s + (0.467 − 0.809i)5-s + (0.982 − 0.183i)7-s + 0.289i·8-s + (−1.19 − 0.692i)10-s + (−0.843 + 0.486i)11-s + (−0.745 + 0.430i)13-s + (−0.272 − 1.45i)14-s − 0.766·16-s + (−0.197 + 0.342i)17-s + (1.09 − 0.634i)19-s + (−0.558 + 0.967i)20-s + (0.721 + 1.24i)22-s + (0.209 + 0.121i)23-s + ⋯

Functional equation

Λ(s)=(189s/2 ΓC(s) L(s)=((−0.749+0.662i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(189s/2 ΓC(s+1/2) L(s)=((−0.749+0.662i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 189189    =    33⋅73^{3} \cdot 7
Sign: −0.749+0.662i-0.749 + 0.662i
Analytic conductor: 1.509171.50917
Root analytic conductor: 1.228481.22848
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ189(143,⋅)\chi_{189} (143, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 189, ( :1/2), −0.749+0.662i)(2,\ 189,\ (\ :1/2),\ -0.749 + 0.662i)

Particular Values

L(1)L(1) ≈\approx 0.445835−1.17786i0.445835 - 1.17786i
L(12)L(\frac12) ≈\approx 0.445835−1.17786i0.445835 - 1.17786i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
7 1+(−2.60+0.486i)T 1 + (-2.60 + 0.486i)T
good2 1+2.09iT−2T2 1 + 2.09iT - 2T^{2}
5 1+(−1.04+1.80i)T+(−2.5−4.33i)T2 1 + (-1.04 + 1.80i)T + (-2.5 - 4.33i)T^{2}
11 1+(2.79−1.61i)T+(5.5−9.52i)T2 1 + (2.79 - 1.61i)T + (5.5 - 9.52i)T^{2}
13 1+(2.68−1.55i)T+(6.5−11.2i)T2 1 + (2.68 - 1.55i)T + (6.5 - 11.2i)T^{2}
17 1+(0.816−1.41i)T+(−8.5−14.7i)T2 1 + (0.816 - 1.41i)T + (-8.5 - 14.7i)T^{2}
19 1+(−4.79+2.76i)T+(9.5−16.4i)T2 1 + (-4.79 + 2.76i)T + (9.5 - 16.4i)T^{2}
23 1+(−1.00−0.580i)T+(11.5+19.9i)T2 1 + (-1.00 - 0.580i)T + (11.5 + 19.9i)T^{2}
29 1+(−7.05−4.07i)T+(14.5+25.1i)T2 1 + (-7.05 - 4.07i)T + (14.5 + 25.1i)T^{2}
31 1+5.96iT−31T2 1 + 5.96iT - 31T^{2}
37 1+(−2.82−4.89i)T+(−18.5+32.0i)T2 1 + (-2.82 - 4.89i)T + (-18.5 + 32.0i)T^{2}
41 1+(1.35+2.34i)T+(−20.5+35.5i)T2 1 + (1.35 + 2.34i)T + (-20.5 + 35.5i)T^{2}
43 1+(0.974−1.68i)T+(−21.5−37.2i)T2 1 + (0.974 - 1.68i)T + (-21.5 - 37.2i)T^{2}
47 1+8.13T+47T2 1 + 8.13T + 47T^{2}
53 1+(−5.27−3.04i)T+(26.5+45.8i)T2 1 + (-5.27 - 3.04i)T + (26.5 + 45.8i)T^{2}
59 1−3.96T+59T2 1 - 3.96T + 59T^{2}
61 1−4.79iT−61T2 1 - 4.79iT - 61T^{2}
67 1+0.673T+67T2 1 + 0.673T + 67T^{2}
71 1−7.01iT−71T2 1 - 7.01iT - 71T^{2}
73 1+(2.96+1.71i)T+(36.5+63.2i)T2 1 + (2.96 + 1.71i)T + (36.5 + 63.2i)T^{2}
79 1+14.1T+79T2 1 + 14.1T + 79T^{2}
83 1+(−1.54+2.67i)T+(−41.5−71.8i)T2 1 + (-1.54 + 2.67i)T + (-41.5 - 71.8i)T^{2}
89 1+(2.45+4.25i)T+(−44.5+77.0i)T2 1 + (2.45 + 4.25i)T + (-44.5 + 77.0i)T^{2}
97 1+(2.07+1.20i)T+(48.5+84.0i)T2 1 + (2.07 + 1.20i)T + (48.5 + 84.0i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−12.03451114114315860776524155473, −11.34274812008040481118658796332, −10.28417760775648183114612332872, −9.536922917017560998927131652154, −8.503050981494284095162200129509, −7.19234877425529491462492095542, −5.18010831137235352432774634915, −4.47907152117764938275520614692, −2.65687486688240048697128031327, −1.36204481387660754923117448507, 2.67116898504546692365944055809, 4.88592622808414784242912477523, 5.64700415049810561167061765936, 6.79934970092097712344464255571, 7.74887611743675391720947098930, 8.458006948627170622249291502005, 9.868237459701506397128339184824, 10.86615629189624096585124751841, 11.96588635213837141393666992112, 13.45107287835890323355474756383

Graph of the ZZ-function along the critical line