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Properties

Label 2-189-63.38-c3-0-8
Degree $2$
Conductor $189$
Sign $0.368 - 0.929i$
Analytic cond. $11.1513$
Root an. cond. $3.33936$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.10i·2-s + 6.77·4-s + (−6.23 + 10.8i)5-s + (11.7 + 14.3i)7-s − 16.3i·8-s + (11.9 + 6.90i)10-s + (−3.22 + 1.86i)11-s + (−68.0 + 39.2i)13-s + (15.8 − 12.9i)14-s + 36.0·16-s + (−56.8 + 98.4i)17-s + (33.4 − 19.3i)19-s + (−42.2 + 73.1i)20-s + (2.05 + 3.56i)22-s + (32.5 + 18.8i)23-s + ⋯
L(s)  = 1  − 0.391i·2-s + 0.846·4-s + (−0.557 + 0.966i)5-s + (0.632 + 0.774i)7-s − 0.722i·8-s + (0.378 + 0.218i)10-s + (−0.0883 + 0.0510i)11-s + (−1.45 + 0.838i)13-s + (0.303 − 0.247i)14-s + 0.563·16-s + (−0.810 + 1.40i)17-s + (0.403 − 0.233i)19-s + (−0.472 + 0.818i)20-s + (0.0199 + 0.0345i)22-s + (0.295 + 0.170i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.368 - 0.929i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.368 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.368 - 0.929i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ 0.368 - 0.929i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.43108 + 0.971978i\)
\(L(\frac12)\) \(\approx\) \(1.43108 + 0.971978i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-11.7 - 14.3i)T \)
good2 \( 1 + 1.10iT - 8T^{2} \)
5 \( 1 + (6.23 - 10.8i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (3.22 - 1.86i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (68.0 - 39.2i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (56.8 - 98.4i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-33.4 + 19.3i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-32.5 - 18.8i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-144. - 83.7i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 212. iT - 2.97e4T^{2} \)
37 \( 1 + (-132. - 228. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-190. - 329. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-90.2 + 156. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 154.T + 1.03e5T^{2} \)
53 \( 1 + (162. + 93.5i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 - 468.T + 2.05e5T^{2} \)
61 \( 1 + 356. iT - 2.26e5T^{2} \)
67 \( 1 + 96.7T + 3.00e5T^{2} \)
71 \( 1 + 705. iT - 3.57e5T^{2} \)
73 \( 1 + (631. + 364. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + 409.T + 4.93e5T^{2} \)
83 \( 1 + (-321. + 556. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (74.4 + 128. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-605. - 349. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−11.92881242484591234592129220201, −11.42410407006304087100951951598, −10.61015736129051819820707707378, −9.506243219940851289591291526986, −8.065739984622307735005625605389, −7.12416929107584894910876988003, −6.21018498749083712826577169677, −4.55292393716619303943130286360, −2.98210654930364764941131468440, −1.97569664702744405288619477734, 0.74340645993883989253204749440, 2.63159888916578290010624230079, 4.53035719259519685308487680322, 5.34893988725404128717528178660, 7.06403477121684634428338655962, 7.63581789511107605466023726657, 8.624435951709803582058332324106, 10.01924689380349021902523089574, 11.07373952534824895129701950814, 11.93687163358503834319560516830

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