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Properties

Label 2-189-63.61-c2-0-10
Degree $2$
Conductor $189$
Sign $-0.903 + 0.428i$
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  + (−0.902 − 1.56i)2-s + (0.371 − 0.643i)4-s − 5.75i·5-s + (6.44 − 2.73i)7-s − 8.56·8-s + (−8.99 + 5.19i)10-s + 10.1·11-s + (−9.76 + 5.63i)13-s + (−10.0 − 7.59i)14-s + (6.23 + 10.8i)16-s + (15.3 − 8.86i)17-s + (−26.6 − 15.3i)19-s + (−3.70 − 2.13i)20-s + (−9.19 − 15.9i)22-s − 31.3·23-s + ⋯
L(s)  = 1  + (−0.451 − 0.781i)2-s + (0.0929 − 0.160i)4-s − 1.15i·5-s + (0.920 − 0.390i)7-s − 1.07·8-s + (−0.899 + 0.519i)10-s + 0.926·11-s + (−0.750 + 0.433i)13-s + (−0.720 − 0.542i)14-s + (0.389 + 0.675i)16-s + (0.903 − 0.521i)17-s + (−1.40 − 0.808i)19-s + (−0.185 − 0.106i)20-s + (−0.417 − 0.723i)22-s − 1.36·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.903 + 0.428i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.903 + 0.428i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.903 + 0.428i$
Analytic conductor: \(5.14987\)
Root analytic conductor: \(2.26933\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1),\ -0.903 + 0.428i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.266149 - 1.18239i\)
\(L(\frac12)\) \(\approx\) \(0.266149 - 1.18239i\)
\(L(2)\) 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 \( 1 + (-6.44 + 2.73i)T \)
good2 \( 1 + (0.902 + 1.56i)T + (-2 + 3.46i)T^{2} \)
5 \( 1 + 5.75iT - 25T^{2} \)
11 \( 1 - 10.1T + 121T^{2} \)
13 \( 1 + (9.76 - 5.63i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (-15.3 + 8.86i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (26.6 + 15.3i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + 31.3T + 529T^{2} \)
29 \( 1 + (1.48 - 2.56i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-25.0 - 14.4i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (11.0 - 19.1i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-9.95 + 5.74i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-19.6 + 34.0i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-18.5 + 10.7i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-22.8 - 39.6i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (11.1 + 6.42i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-72.4 + 41.8i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (10.1 - 17.6i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 92.6T + 5.04e3T^{2} \)
73 \( 1 + (-88.5 + 51.0i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-50.2 - 86.9i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-20.6 - 11.9i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (48.4 + 27.9i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (69.6 + 40.1i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−11.98388896941425765344483044904, −10.96058104866046301429376924883, −9.928018538226633795215026020124, −9.048880166055800834967934706140, −8.212880042954682794093828603865, −6.71946662456643684505603503578, −5.26005867231045533814726128701, −4.16883593833993784131257795159, −2.09183899673013468407234506093, −0.843703988648583806316609633107, 2.35113678887919124412725973911, 3.89392059754450620181846209925, 5.79826807040470921541486291355, 6.61727094332969504806025759645, 7.75976055951881875203342799993, 8.342309283224268582515377234951, 9.707178760783606243176076736236, 10.73561309236181270570359169322, 11.83071062572255973453135781274, 12.46214980233535870536980746411

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