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Properties

Label 2-42e2-9.4-c1-0-8
Degree $2$
Conductor $1764$
Sign $0.449 - 0.893i$
Analytic cond. $14.0856$
Root an. cond. $3.75308$
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  + (−1.64 − 0.545i)3-s + (−0.849 + 1.47i)5-s + (2.40 + 1.79i)9-s + (−1.23 − 2.14i)11-s + (0.388 − 0.673i)13-s + (2.19 − 1.95i)15-s − 2.81·17-s + 4.98·19-s + (−0.356 + 0.616i)23-s + (1.05 + 1.82i)25-s + (−2.97 − 4.25i)27-s + (−2.25 − 3.90i)29-s + (2.54 − 4.41i)31-s + (0.866 + 4.20i)33-s − 6.87·37-s + ⋯
L(s)  = 1  + (−0.949 − 0.314i)3-s + (−0.380 + 0.658i)5-s + (0.801 + 0.597i)9-s + (−0.373 − 0.646i)11-s + (0.107 − 0.186i)13-s + (0.567 − 0.505i)15-s − 0.681·17-s + 1.14·19-s + (−0.0742 + 0.128i)23-s + (0.211 + 0.365i)25-s + (−0.572 − 0.819i)27-s + (−0.418 − 0.725i)29-s + (0.457 − 0.793i)31-s + (0.150 + 0.731i)33-s − 1.13·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.449 - 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.449 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.449 - 0.893i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 0.449 - 0.893i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8639534433\)
\(L(\frac12)\) \(\approx\) \(0.8639534433\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.64 + 0.545i)T \)
7 \( 1 \)
good5 \( 1 + (0.849 - 1.47i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.23 + 2.14i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.388 + 0.673i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 2.81T + 17T^{2} \)
19 \( 1 - 4.98T + 19T^{2} \)
23 \( 1 + (0.356 - 0.616i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.25 + 3.90i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.54 + 4.41i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6.87T + 37T^{2} \)
41 \( 1 + (2.93 - 5.08i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.32 - 4.03i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.49 - 11.2i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 1.88T + 53T^{2} \)
59 \( 1 + (-7.14 + 12.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.15 - 12.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.99 - 6.91i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.2T + 71T^{2} \)
73 \( 1 + 4.98T + 73T^{2} \)
79 \( 1 + (-4.60 - 7.97i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.40 - 7.63i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 9.65T + 89T^{2} \)
97 \( 1 + (-4.32 - 7.48i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.628862336306115823873257152184, −8.507963438155097461852983914750, −7.61477381262671285275471645510, −7.10980379452563710484960839287, −6.16805722380876958658588015082, −5.55580178916776948421425464961, −4.59111320660843624605352238204, −3.54131843844701919339776408986, −2.47585520508381724223350860315, −0.979330793066210489694237825233, 0.46937208633514497947741358456, 1.82660533984197579320146263773, 3.42032787559572487649908137769, 4.39442907023558392510257909643, 5.04930718727211377081973311919, 5.71327199690212884772058246978, 6.91642658223995165976354794515, 7.30008996912562706906628100996, 8.581642160168505375243193099629, 9.082878253492080175524118103767

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