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Properties

Label 2-189-189.25-c1-0-6
Degree $2$
Conductor $189$
Sign $0.584 - 0.811i$
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  + (−2.11 + 0.771i)2-s + (1.70 + 0.287i)3-s + (2.36 − 1.98i)4-s + (−1.51 − 0.552i)5-s + (−3.84 + 0.707i)6-s + (2.02 + 1.70i)7-s + (−1.22 + 2.11i)8-s + (2.83 + 0.983i)9-s + 3.64·10-s + (0.906 − 0.329i)11-s + (4.60 − 2.70i)12-s + (−0.0325 − 0.184i)13-s + (−5.60 − 2.04i)14-s + (−2.43 − 1.38i)15-s + (−0.115 + 0.652i)16-s + 4.70·17-s + ⋯
L(s)  = 1  + (−1.49 + 0.545i)2-s + (0.986 + 0.166i)3-s + (1.18 − 0.990i)4-s + (−0.678 − 0.247i)5-s + (−1.56 + 0.288i)6-s + (0.764 + 0.644i)7-s + (−0.431 + 0.747i)8-s + (0.944 + 0.327i)9-s + 1.15·10-s + (0.273 − 0.0994i)11-s + (1.32 − 0.780i)12-s + (−0.00901 − 0.0511i)13-s + (−1.49 − 0.547i)14-s + (−0.628 − 0.356i)15-s + (−0.0287 + 0.163i)16-s + 1.14·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.584 - 0.811i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.584 - 0.811i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.715444 + 0.366485i\)
\(L(\frac12)\) \(\approx\) \(0.715444 + 0.366485i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.70 - 0.287i)T \)
7 \( 1 + (-2.02 - 1.70i)T \)
good2 \( 1 + (2.11 - 0.771i)T + (1.53 - 1.28i)T^{2} \)
5 \( 1 + (1.51 + 0.552i)T + (3.83 + 3.21i)T^{2} \)
11 \( 1 + (-0.906 + 0.329i)T + (8.42 - 7.07i)T^{2} \)
13 \( 1 + (0.0325 + 0.184i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 - 4.70T + 17T^{2} \)
19 \( 1 - 0.504T + 19T^{2} \)
23 \( 1 + (-1.36 - 7.72i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (1.02 - 5.82i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (2.85 - 2.39i)T + (5.38 - 30.5i)T^{2} \)
37 \( 1 + (-4.84 + 8.38i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.93 + 10.9i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (6.60 + 5.54i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (1.15 + 0.972i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (-1.38 + 2.40i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.120 + 0.685i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (6.80 + 5.71i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-2.99 - 1.09i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (1.43 + 2.49i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.0174 - 0.0301i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.85 - 2.85i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-2.38 + 13.5i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + 11.4T + 89T^{2} \)
97 \( 1 + (7.73 + 6.48i)T + (16.8 + 95.5i)T^{2} \)
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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

−12.57507436406728291638845234029, −11.48062146937029772384972626156, −10.37084483179088452108635460531, −9.303307117216222014146831567286, −8.701939969893617220009276793025, −7.82161068517866624028879845499, −7.25555355117796960872618372938, −5.42391171701444074253336641062, −3.66808713750857484794962816990, −1.66134090742568107743088151124, 1.32114358756247533191997263321, 2.91945574675749327620093584416, 4.35445658103201640603904487979, 6.87618383683391921066632808393, 7.998450723882681284453953376544, 8.106683893705571622601139670560, 9.492831295448653875093392058935, 10.19406761604216464890961573753, 11.25969623013716253667721095485, 12.01220493024894088129775544553

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