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Properties

Label 2-189-21.5-c1-0-2
Degree $2$
Conductor $189$
Sign $0.997 - 0.0633i$
Analytic cond. $1.50917$
Root an. cond. $1.22848$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 − 0.707i)2-s + (−1.22 + 2.12i)5-s + (2.5 + 0.866i)7-s + 2.82i·8-s + (3 − 1.73i)10-s + (4.89 − 2.82i)11-s + 3.46i·13-s + (−2.44 − 2.82i)14-s + (2.00 − 3.46i)16-s + (−1.22 − 2.12i)17-s + (1.5 + 0.866i)19-s − 8·22-s + (6.12 + 3.53i)23-s + (−0.499 − 0.866i)25-s + (2.44 − 4.24i)26-s + ⋯
L(s)  = 1  + (−0.866 − 0.499i)2-s + (−0.547 + 0.948i)5-s + (0.944 + 0.327i)7-s + 0.999i·8-s + (0.948 − 0.547i)10-s + (1.47 − 0.852i)11-s + 0.960i·13-s + (−0.654 − 0.755i)14-s + (0.500 − 0.866i)16-s + (−0.297 − 0.514i)17-s + (0.344 + 0.198i)19-s − 1.70·22-s + (1.27 + 0.737i)23-s + (−0.0999 − 0.173i)25-s + (0.480 − 0.832i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.997 - 0.0633i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.997 - 0.0633i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.759880 + 0.0240868i\)
\(L(\frac12)\) \(\approx\) \(0.759880 + 0.0240868i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 - 0.866i)T \)
good2 \( 1 + (1.22 + 0.707i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (1.22 - 2.12i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.89 + 2.82i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 + (1.22 + 2.12i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.5 - 0.866i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.12 - 3.53i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.41iT - 29T^{2} \)
31 \( 1 + (4.5 - 2.59i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.44T + 41T^{2} \)
43 \( 1 + 7T + 43T^{2} \)
47 \( 1 + (3.67 - 6.36i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.22 + 0.707i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (7.34 + 12.7i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.5 + 2.59i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.82iT - 71T^{2} \)
73 \( 1 + (-10.5 + 6.06i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 9.79T + 83T^{2} \)
89 \( 1 + (-1.22 + 2.12i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.66iT - 97T^{2} \)
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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

−11.96400562116509780620600550302, −11.19457471147414856829363711052, −11.02894770126410532829121563020, −9.391616437914352980891772129110, −8.876555614053637804120198106225, −7.65998398712494571806786748335, −6.52859308162620839699364821736, −4.99544349149552491414040248634, −3.36655132950682273292083537907, −1.58810241130337706455791704347, 1.11851154836978925579362295626, 3.93012448432945952561673779543, 4.88792383534151970113164416805, 6.70539426307136703355588343480, 7.68868794836723936737761501634, 8.519379868362428858343814575190, 9.196388511497976356199062169067, 10.38069074111549190397833312242, 11.65055438885056191719544784583, 12.48791932276266308273956140612

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