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Properties

Label 2-189-7.6-c2-0-18
Degree $2$
Conductor $189$
Sign $0.947 + 0.320i$
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  + 3.35·2-s + 7.24·4-s − 2.40i·5-s + (2.24 − 6.63i)7-s + 10.8·8-s − 8.06i·10-s + 3.35·11-s + 16.3i·13-s + (7.51 − 22.2i)14-s + 7.48·16-s + 24.6i·17-s − 2.15i·19-s − 17.4i·20-s + 11.2·22-s − 33.4·23-s + ⋯
L(s)  = 1  + 1.67·2-s + 1.81·4-s − 0.481i·5-s + (0.320 − 0.947i)7-s + 1.35·8-s − 0.806i·10-s + 0.304·11-s + 1.25i·13-s + (0.537 − 1.58i)14-s + 0.467·16-s + 1.44i·17-s − 0.113i·19-s − 0.871i·20-s + 0.511·22-s − 1.45·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.320i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.947 + 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.947 + 0.320i$
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: \((2,\ 189,\ (\ :1),\ 0.947 + 0.320i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.64962 - 0.600453i\)
\(L(\frac12)\) \(\approx\) \(3.64962 - 0.600453i\)
\(L(2)\) 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 \( 1 + (-2.24 + 6.63i)T \)
good2 \( 1 - 3.35T + 4T^{2} \)
5 \( 1 + 2.40iT - 25T^{2} \)
11 \( 1 - 3.35T + 121T^{2} \)
13 \( 1 - 16.3iT - 169T^{2} \)
17 \( 1 - 24.6iT - 289T^{2} \)
19 \( 1 + 2.15iT - 361T^{2} \)
23 \( 1 + 33.4T + 529T^{2} \)
29 \( 1 + 41.0T + 841T^{2} \)
31 \( 1 + 32.9iT - 961T^{2} \)
37 \( 1 - 37.4T + 1.36e3T^{2} \)
41 \( 1 + 57.6iT - 1.68e3T^{2} \)
43 \( 1 - 7.69T + 1.84e3T^{2} \)
47 \( 1 - 89.5iT - 2.20e3T^{2} \)
53 \( 1 + 18.4T + 2.80e3T^{2} \)
59 \( 1 + 13.8iT - 3.48e3T^{2} \)
61 \( 1 - 24.7iT - 3.72e3T^{2} \)
67 \( 1 - 99.1T + 4.48e3T^{2} \)
71 \( 1 - 64.7T + 5.04e3T^{2} \)
73 \( 1 - 50.7iT - 5.32e3T^{2} \)
79 \( 1 - 18.2T + 6.24e3T^{2} \)
83 \( 1 + 70.2iT - 6.88e3T^{2} \)
89 \( 1 + 36.0iT - 7.92e3T^{2} \)
97 \( 1 + 143. iT - 9.40e3T^{2} \)
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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

−12.56470960274408691905479802080, −11.55608845601428770775211905142, −10.80210348496776296370010388638, −9.370870025874099525878405724825, −7.903743216637088436198477626505, −6.70985858688784549581227312952, −5.75572737368666075425379270988, −4.33740512830925116492041330836, −3.91960542159447565800315452044, −1.88475467160006019278115505507, 2.42446333807644267983495345902, 3.45055456686813503656736236245, 4.95469738439396432097101514905, 5.72407764929297424826594709291, 6.80049895231973182371176598755, 8.047262313825259384963317720941, 9.541953563347563273654485032933, 10.91478947700738921992257233605, 11.73035676248263532005591320891, 12.45495546747253439540415278406

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