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Properties

Label 2-189-7.2-c1-0-2
Degree $2$
Conductor $189$
Sign $-0.574 - 0.818i$
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  + (−0.730 + 1.26i)2-s + (−0.0665 − 0.115i)4-s + (−0.296 + 0.514i)5-s + (2.32 + 1.26i)7-s − 2.72·8-s + (−0.433 − 0.750i)10-s + (2.23 + 3.86i)11-s − 4.51·13-s + (−3.29 + 2.01i)14-s + (2.12 − 3.67i)16-s + (−0.136 − 0.236i)17-s + (−1.43 + 2.48i)19-s + 0.0789·20-s − 6.51·22-s + (2.52 − 4.37i)23-s + ⋯
L(s)  = 1  + (−0.516 + 0.894i)2-s + (−0.0332 − 0.0576i)4-s + (−0.132 + 0.229i)5-s + (0.878 + 0.478i)7-s − 0.964·8-s + (−0.137 − 0.237i)10-s + (0.672 + 1.16i)11-s − 1.25·13-s + (−0.881 + 0.538i)14-s + (0.531 − 0.919i)16-s + (−0.0331 − 0.0574i)17-s + (−0.328 + 0.569i)19-s + 0.0176·20-s − 1.38·22-s + (0.526 − 0.912i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.574 - 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.574 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.574 - 0.818i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ -0.574 - 0.818i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.424858 + 0.817403i\)
\(L(\frac12)\) \(\approx\) \(0.424858 + 0.817403i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.32 - 1.26i)T \)
good2 \( 1 + (0.730 - 1.26i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (0.296 - 0.514i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.23 - 3.86i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.51T + 13T^{2} \)
17 \( 1 + (0.136 + 0.236i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.43 - 2.48i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.52 + 4.37i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.352T + 29T^{2} \)
31 \( 1 + (1.25 + 2.17i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.32 + 5.75i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 10.8T + 41T^{2} \)
43 \( 1 - 3.38T + 43T^{2} \)
47 \( 1 + (-6.21 + 10.7i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.66 - 9.80i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.02 + 6.97i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.36 + 2.36i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.93 + 5.08i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.60T + 71T^{2} \)
73 \( 1 + (5.55 + 9.62i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.58 - 9.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 16.5T + 83T^{2} \)
89 \( 1 + (2.68 - 4.65i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 2.26T + 97T^{2} \)
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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

−12.56915864459932393203259331739, −12.10154013492408082479071708865, −10.94575917784581109431827338093, −9.579718292438336800895305307580, −8.803964799284291286266773118524, −7.59681237389984299935567103570, −7.08073980977407271897873381889, −5.72347437717030477254089544245, −4.40244857594651719507545620912, −2.40491511976131716383366971973, 1.05382292861248251908236924670, 2.75295905682635696080946726110, 4.38581712729720874802127500090, 5.79672416169981861450977037760, 7.22550737500618431112605846810, 8.496230878752029202630775541281, 9.316898687775732819777068258768, 10.42594738686001698543445670507, 11.25464412325648791283986621158, 11.83399094579787224312406227773

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