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Properties

Label 2-189-189.5-c1-0-11
Degree $2$
Conductor $189$
Sign $0.679 - 0.733i$
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  + (1.95 − 0.345i)2-s + (−1.39 + 1.02i)3-s + (1.83 − 0.667i)4-s + (−0.649 + 3.68i)5-s + (−2.37 + 2.49i)6-s + (2.58 − 0.552i)7-s + (−0.0842 + 0.0486i)8-s + (0.891 − 2.86i)9-s + 7.43i·10-s + (0.526 − 0.0928i)11-s + (−1.87 + 2.81i)12-s + (3.22 − 3.84i)13-s + (4.87 − 1.97i)14-s + (−2.87 − 5.80i)15-s + (−3.13 + 2.63i)16-s − 4.53·17-s + ⋯
L(s)  = 1  + (1.38 − 0.244i)2-s + (−0.805 + 0.592i)3-s + (0.916 − 0.333i)4-s + (−0.290 + 1.64i)5-s + (−0.970 + 1.01i)6-s + (0.977 − 0.208i)7-s + (−0.0297 + 0.0171i)8-s + (0.297 − 0.954i)9-s + 2.35i·10-s + (0.158 − 0.0280i)11-s + (−0.540 + 0.812i)12-s + (0.894 − 1.06i)13-s + (1.30 − 0.527i)14-s + (−0.742 − 1.49i)15-s + (−0.784 + 0.658i)16-s − 1.10·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.679 - 0.733i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.679 - 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.679 - 0.733i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.679 - 0.733i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67845 + 0.733119i\)
\(L(\frac12)\) \(\approx\) \(1.67845 + 0.733119i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.39 - 1.02i)T \)
7 \( 1 + (-2.58 + 0.552i)T \)
good2 \( 1 + (-1.95 + 0.345i)T + (1.87 - 0.684i)T^{2} \)
5 \( 1 + (0.649 - 3.68i)T + (-4.69 - 1.71i)T^{2} \)
11 \( 1 + (-0.526 + 0.0928i)T + (10.3 - 3.76i)T^{2} \)
13 \( 1 + (-3.22 + 3.84i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + 4.53T + 17T^{2} \)
19 \( 1 + 1.20iT - 19T^{2} \)
23 \( 1 + (-5.35 + 6.37i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (-2.30 - 2.75i)T + (-5.03 + 28.5i)T^{2} \)
31 \( 1 + (0.0334 + 0.0920i)T + (-23.7 + 19.9i)T^{2} \)
37 \( 1 + (0.267 + 0.463i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.79 + 4.85i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (0.0316 + 0.0115i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-5.19 - 1.88i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (8.57 - 4.95i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.583 - 0.489i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-2.21 + 6.08i)T + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (1.14 - 6.48i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-3.13 - 1.80i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.76 + 3.32i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.909 + 5.15i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (1.39 - 1.16i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + 9.21T + 89T^{2} \)
97 \( 1 + (4.37 - 12.0i)T + (-74.3 - 62.3i)T^{2} \)
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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

−12.64002798308935370244369479492, −11.55347890219902373874552343884, −10.84718758816533908074842536149, −10.62996727545444563232872481578, −8.668080393374953276293076636931, −6.99196275252002892636718298302, −6.19536102595189021127701849736, −5.00537260043365436046636038293, −3.97269548795570768099179771681, −2.87481072522384202088931738819, 1.54346126567714391196270670820, 4.22314541514449113764600441404, 4.88216914289051433189769410904, 5.72931026493107742735202246866, 6.89666078928681928103056481542, 8.262792134062950332870449504144, 9.194810595198723040598192455965, 11.32954548052963590393516753629, 11.65917607017913961268209899644, 12.57346584960623238992983206248

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