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Properties

Label 2-42e2-63.20-c1-0-24
Degree $2$
Conductor $1764$
Sign $0.392 + 0.919i$
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  + (−0.240 + 1.71i)3-s + (−1.48 − 2.57i)5-s + (−2.88 − 0.825i)9-s + (4.09 + 2.36i)11-s + (−3.54 + 2.04i)13-s + (4.76 − 1.92i)15-s + 1.67·17-s − 4.91i·19-s + (−4.25 + 2.45i)23-s + (−1.91 + 3.30i)25-s + (2.11 − 4.74i)27-s + (0.238 + 0.137i)29-s + (−1.38 + 0.801i)31-s + (−5.04 + 6.45i)33-s + 3.39·37-s + ⋯
L(s)  = 1  + (−0.138 + 0.990i)3-s + (−0.664 − 1.15i)5-s + (−0.961 − 0.275i)9-s + (1.23 + 0.712i)11-s + (−0.981 + 0.566i)13-s + (1.23 − 0.497i)15-s + 0.405·17-s − 1.12i·19-s + (−0.886 + 0.511i)23-s + (−0.382 + 0.661i)25-s + (0.406 − 0.913i)27-s + (0.0442 + 0.0255i)29-s + (−0.249 + 0.143i)31-s + (−0.877 + 1.12i)33-s + 0.557·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.392 + 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.392 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.392 + 0.919i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 0.392 + 0.919i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9385704199\)
\(L(\frac12)\) \(\approx\) \(0.9385704199\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.240 - 1.71i)T \)
7 \( 1 \)
good5 \( 1 + (1.48 + 2.57i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-4.09 - 2.36i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.54 - 2.04i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.67T + 17T^{2} \)
19 \( 1 + 4.91iT - 19T^{2} \)
23 \( 1 + (4.25 - 2.45i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.238 - 0.137i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.38 - 0.801i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.39T + 37T^{2} \)
41 \( 1 + (3.55 + 6.15i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.22 + 9.05i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.49 + 9.52i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 0.816iT - 53T^{2} \)
59 \( 1 + (1.37 + 2.38i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.23 + 3.60i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.80 + 10.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.4iT - 71T^{2} \)
73 \( 1 + 15.7iT - 73T^{2} \)
79 \( 1 + (-6.15 + 10.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.03 - 6.99i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 9.21T + 89T^{2} \)
97 \( 1 + (-7.00 - 4.04i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.164677647923456107343035551935, −8.694515141951417602257230685443, −7.62102301508679354368111941738, −6.82524953029354194082526092990, −5.65513465756713094903086719654, −4.79804806740777164878401427103, −4.31010361729151710768825860862, −3.54601240812612022285633089535, −2.01941231306542333131584642426, −0.39764624073231407532316243772, 1.18350002422668485050308418688, 2.58704129833304331336575673680, 3.32315424664563865040279681647, 4.34216937423171622482388831812, 5.84857657103611457205959348726, 6.23145770491300288312353541967, 7.16789630821649072585618313600, 7.73832657655239014342866760552, 8.333018700049192708038775610717, 9.450020229260483257858002639091

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