Properties

Label 2-189-189.5-c1-0-5
Degree $2$
Conductor $189$
Sign $0.137 - 0.990i$
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  + (−2.51 + 0.443i)2-s + (−1.35 + 1.07i)3-s + (4.25 − 1.54i)4-s + (−0.181 + 1.03i)5-s + (2.93 − 3.30i)6-s + (2.00 − 1.73i)7-s + (−5.59 + 3.23i)8-s + (0.685 − 2.92i)9-s − 2.67i·10-s + (4.05 − 0.714i)11-s + (−4.11 + 6.68i)12-s + (−2.92 + 3.48i)13-s + (−4.26 + 5.24i)14-s + (−0.862 − 1.59i)15-s + (5.70 − 4.79i)16-s + 3.48·17-s + ⋯
L(s)  = 1  + (−1.77 + 0.313i)2-s + (−0.783 + 0.621i)3-s + (2.12 − 0.774i)4-s + (−0.0813 + 0.461i)5-s + (1.19 − 1.35i)6-s + (0.756 − 0.654i)7-s + (−1.97 + 1.14i)8-s + (0.228 − 0.973i)9-s − 0.846i·10-s + (1.22 − 0.215i)11-s + (−1.18 + 1.92i)12-s + (−0.812 + 0.967i)13-s + (−1.14 + 1.40i)14-s + (−0.222 − 0.411i)15-s + (1.42 − 1.19i)16-s + 0.844·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.137 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.137 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.137 - 0.990i$
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: \((2,\ 189,\ (\ :1/2),\ 0.137 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.333879 + 0.290777i\)
\(L(\frac12)\) \(\approx\) \(0.333879 + 0.290777i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.35 - 1.07i)T \)
7 \( 1 + (-2.00 + 1.73i)T \)
good2 \( 1 + (2.51 - 0.443i)T + (1.87 - 0.684i)T^{2} \)
5 \( 1 + (0.181 - 1.03i)T + (-4.69 - 1.71i)T^{2} \)
11 \( 1 + (-4.05 + 0.714i)T + (10.3 - 3.76i)T^{2} \)
13 \( 1 + (2.92 - 3.48i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 - 3.48T + 17T^{2} \)
19 \( 1 - 5.54iT - 19T^{2} \)
23 \( 1 + (3.90 - 4.65i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (1.57 + 1.87i)T + (-5.03 + 28.5i)T^{2} \)
31 \( 1 + (-0.362 - 0.996i)T + (-23.7 + 19.9i)T^{2} \)
37 \( 1 + (1.64 + 2.84i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.164 + 0.138i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-9.06 - 3.29i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (0.827 + 0.301i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (0.554 - 0.320i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-8.17 - 6.85i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (0.694 - 1.90i)T + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (1.96 - 11.1i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-9.48 - 5.47i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (9.95 + 5.74i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.940 + 5.33i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-7.87 + 6.60i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 - 4.37T + 89T^{2} \)
97 \( 1 + (-3.53 + 9.72i)T + (-74.3 - 62.3i)T^{2} \)
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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

−11.96256406806759652249840016877, −11.52151653304468912110485192442, −10.53950492306962375513503136717, −9.846838062079648836876857693675, −9.016788376791342558901848468160, −7.68058563882883238727824396218, −6.90461450879140725578877271351, −5.80166135508488860313910485812, −3.99439374036050514416663016872, −1.40705504410085777065851497776, 0.890807109318801934193029539743, 2.32525301751620627687646854169, 5.02341158472624526589199229077, 6.46705197167875778711632753711, 7.52506549635264076393529202507, 8.347802492808743318038538846029, 9.277844014666918542397925765830, 10.37744029028815260732640390434, 11.27285486287756583099204433136, 12.10701663740052839898697557993

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