Properties

Degree 2
Conductor $ 3^{3} \cdot 7 $
Sign $0.811 + 0.583i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.34 − 2.33i)2-s + (−2.64 + 4.57i)4-s + (0.794 + 1.37i)5-s + (1.23 + 2.33i)7-s + 8.87·8-s + (2.14 − 3.71i)10-s + (−0.150 + 0.260i)11-s + 2.81·13-s + (3.79 − 6.05i)14-s + (−6.69 − 11.5i)16-s + (−2.93 + 5.08i)17-s + (1.14 + 1.98i)19-s − 8.39·20-s + 0.810·22-s + (−0.944 − 1.63i)23-s + ⋯
L(s)  = 1  + (−0.954 − 1.65i)2-s + (−1.32 + 2.28i)4-s + (0.355 + 0.615i)5-s + (0.468 + 0.883i)7-s + 3.13·8-s + (0.677 − 1.17i)10-s + (−0.0452 + 0.0784i)11-s + 0.779·13-s + (1.01 − 1.61i)14-s + (−1.67 − 2.89i)16-s + (−0.712 + 1.23i)17-s + (0.262 + 0.454i)19-s − 1.87·20-s + 0.172·22-s + (−0.196 − 0.341i)23-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 + 0.583i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.811 + 0.583i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(189\)    =    \(3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.811 + 0.583i$
motivic weight  =  \(1\)
character  :  $\chi_{189} (109, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 189,\ (\ :1/2),\ 0.811 + 0.583i)$
$L(1)$  $\approx$  $0.708443 - 0.228263i$
$L(\frac12)$  $\approx$  $0.708443 - 0.228263i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{3,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
7 \( 1 + (-1.23 - 2.33i)T \)
good2 \( 1 + (1.34 + 2.33i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-0.794 - 1.37i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.150 - 0.260i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.81T + 13T^{2} \)
17 \( 1 + (2.93 - 5.08i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.14 - 1.98i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.944 + 1.63i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.52T + 29T^{2} \)
31 \( 1 + (-2.40 + 4.16i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.23 - 3.87i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.90T + 41T^{2} \)
43 \( 1 + 9.09T + 43T^{2} \)
47 \( 1 + (-1.60 - 2.78i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.00 - 1.74i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.44 + 4.23i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.78 + 6.56i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.356 - 0.616i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.8T + 71T^{2} \)
73 \( 1 + (-5.83 + 10.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.833 + 1.44i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.43T + 83T^{2} \)
89 \( 1 + (-4.67 - 8.10i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 12.5T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−12.18565467217350042756141095043, −11.26548095522022143031985555182, −10.61164174659470080763984009745, −9.712717715651509793069592217661, −8.654633930825879448001257461482, −8.017735535194880030236004534076, −6.22829859181164130538672971329, −4.33233797969404216597051904151, −2.91194813142760716286759211651, −1.77890331276711388954673613819, 1.04657415131580579698762638850, 4.52996705262543845454074689183, 5.45261081915559241856239925822, 6.70481678199360586900680251303, 7.51881026177251543791451318937, 8.582880720002424156430600082358, 9.275443740665464728477409723429, 10.30037411606787747437577679973, 11.32674082806120717454119681401, 13.30030120861463747642552938222

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