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Properties

Label 2-189-7.6-c2-0-5
Degree $2$
Conductor $189$
Sign $0.371 - 0.928i$
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  − 4·4-s + (6.5 + 2.59i)7-s + 25.9i·13-s + 16·16-s + 36.3i·19-s + 25·25-s + (−26 − 10.3i)28-s − 41.5i·31-s − 47·37-s + 22·43-s + (35.5 + 33.7i)49-s − 103. i·52-s + 15.5i·61-s − 64·64-s − 109·67-s + ⋯
L(s)  = 1  − 4-s + (0.928 + 0.371i)7-s + 1.99i·13-s + 16-s + 1.91i·19-s + 25-s + (−0.928 − 0.371i)28-s − 1.34i·31-s − 1.27·37-s + 0.511·43-s + (0.724 + 0.689i)49-s − 1.99i·52-s + 0.255i·61-s − 64-s − 1.62·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.371 - 0.928i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.371 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.371 - 0.928i$
Analytic conductor: \(5.14987\)
Root analytic conductor: \(2.26933\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1),\ 0.371 - 0.928i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.983071 + 0.665754i\)
\(L(\frac12)\) \(\approx\) \(0.983071 + 0.665754i\)
\(L(2)\) 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 \( 1 + (-6.5 - 2.59i)T \)
good2 \( 1 + 4T^{2} \)
5 \( 1 - 25T^{2} \)
11 \( 1 + 121T^{2} \)
13 \( 1 - 25.9iT - 169T^{2} \)
17 \( 1 - 289T^{2} \)
19 \( 1 - 36.3iT - 361T^{2} \)
23 \( 1 + 529T^{2} \)
29 \( 1 + 841T^{2} \)
31 \( 1 + 41.5iT - 961T^{2} \)
37 \( 1 + 47T + 1.36e3T^{2} \)
41 \( 1 - 1.68e3T^{2} \)
43 \( 1 - 22T + 1.84e3T^{2} \)
47 \( 1 - 2.20e3T^{2} \)
53 \( 1 + 2.80e3T^{2} \)
59 \( 1 - 3.48e3T^{2} \)
61 \( 1 - 15.5iT - 3.72e3T^{2} \)
67 \( 1 + 109T + 4.48e3T^{2} \)
71 \( 1 + 5.04e3T^{2} \)
73 \( 1 + 109. iT - 5.32e3T^{2} \)
79 \( 1 - 131T + 6.24e3T^{2} \)
83 \( 1 - 6.88e3T^{2} \)
89 \( 1 - 7.92e3T^{2} \)
97 \( 1 + 98.7iT - 9.40e3T^{2} \)
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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

−12.39468578412390354196596857084, −11.71806896707312518314054765667, −10.49939031211302205708103797508, −9.345521451682793689348305944786, −8.634578279212685524118154523541, −7.60373479067498568667962273143, −6.07476919334701034285955961500, −4.84451595149901926247581337760, −3.91257805696893298484065793291, −1.72130211511069732702418021271, 0.78452085254629715719155717232, 3.14487885448521801597493251481, 4.71065469859008843809431811849, 5.38040556498843869904585049803, 7.16054145908955019311848036993, 8.241270827466330803904759730102, 8.966900277572951362112596132517, 10.31949002103609815991745590462, 10.93626709095079834815450191126, 12.36116892890423398699508567263

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