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Properties

Label 2-189-3.2-c2-0-15
Degree $2$
Conductor $189$
Sign $-i$
Analytic cond. $5.14987$
Root an. cond. $2.26933$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.92i·2-s − 11.4·4-s − 7.06i·5-s − 2.64·7-s + 29.1i·8-s − 27.7·10-s − 4.96i·11-s + 5.02·13-s + 10.3i·14-s + 68.9·16-s + 5.51i·17-s − 18.2·19-s + 80.7i·20-s − 19.5·22-s + 10.6i·23-s + ⋯
L(s)  = 1  − 1.96i·2-s − 2.85·4-s − 1.41i·5-s − 0.377·7-s + 3.64i·8-s − 2.77·10-s − 0.451i·11-s + 0.386·13-s + 0.742i·14-s + 4.30·16-s + 0.324i·17-s − 0.961·19-s + 4.03i·20-s − 0.887·22-s + 0.464i·23-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-i$
Analytic conductor: \(5.14987\)
Root analytic conductor: \(2.26933\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (134, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1),\ -i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.538647 + 0.538647i\)
\(L(\frac12)\) \(\approx\) \(0.538647 + 0.538647i\)
\(L(2)\) 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 \( 1 + 2.64T \)
good2 \( 1 + 3.92iT - 4T^{2} \)
5 \( 1 + 7.06iT - 25T^{2} \)
11 \( 1 + 4.96iT - 121T^{2} \)
13 \( 1 - 5.02T + 169T^{2} \)
17 \( 1 - 5.51iT - 289T^{2} \)
19 \( 1 + 18.2T + 361T^{2} \)
23 \( 1 - 10.6iT - 529T^{2} \)
29 \( 1 + 39.9iT - 841T^{2} \)
31 \( 1 + 2.71T + 961T^{2} \)
37 \( 1 + 13.0T + 1.36e3T^{2} \)
41 \( 1 + 33.1iT - 1.68e3T^{2} \)
43 \( 1 + 61.4T + 1.84e3T^{2} \)
47 \( 1 + 73.8iT - 2.20e3T^{2} \)
53 \( 1 - 5.94iT - 2.80e3T^{2} \)
59 \( 1 - 44.6iT - 3.48e3T^{2} \)
61 \( 1 + 13.3T + 3.72e3T^{2} \)
67 \( 1 + 105.T + 4.48e3T^{2} \)
71 \( 1 + 10.1iT - 5.04e3T^{2} \)
73 \( 1 - 133.T + 5.32e3T^{2} \)
79 \( 1 - 50.4T + 6.24e3T^{2} \)
83 \( 1 + 41.5iT - 6.88e3T^{2} \)
89 \( 1 - 53.3iT - 7.92e3T^{2} \)
97 \( 1 - 97.8T + 9.40e3T^{2} \)
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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

−11.76167739781350625107717884699, −10.69664064481124004475977370257, −9.767855926859392698493969096716, −8.840944808088591777742826353415, −8.281744236232546009845424005662, −5.70082996107911901854815434096, −4.56242662922455167796602471835, −3.57122058266748149423559425346, −1.88366755440864623884734840587, −0.46016566572886150955268687412, 3.42124542233371734198814536990, 4.81493181686830312357725408420, 6.25346784630534494444541687076, 6.73886002824209458229323632330, 7.64781056151194879082916491044, 8.759434392604492251171836964579, 9.829985536764908063799729727966, 10.77074343569310957413250413548, 12.54704818687192883433573607016, 13.48333115110264557221176066668

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