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Properties

Label 2-42e2-9.4-c1-0-37
Degree $2$
Conductor $1764$
Sign $-0.939 - 0.342i$
Analytic cond. $14.0856$
Root an. cond. $3.75308$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 + 0.866i)3-s + (1 − 1.73i)5-s + (1.5 − 2.59i)9-s + (−2 − 3.46i)11-s + (1.5 − 2.59i)13-s + 3.46i·15-s − 7·17-s − 5·19-s + (−2 + 3.46i)23-s + (0.500 + 0.866i)25-s + 5.19i·27-s + (0.5 + 0.866i)29-s + (−1.5 + 2.59i)31-s + (6 + 3.46i)33-s + 11·37-s + ⋯
L(s)  = 1  + (−0.866 + 0.499i)3-s + (0.447 − 0.774i)5-s + (0.5 − 0.866i)9-s + (−0.603 − 1.04i)11-s + (0.416 − 0.720i)13-s + 0.894i·15-s − 1.69·17-s − 1.14·19-s + (−0.417 + 0.722i)23-s + (0.100 + 0.173i)25-s + 0.999i·27-s + (0.0928 + 0.160i)29-s + (−0.269 + 0.466i)31-s + (1.04 + 0.603i)33-s + 1.80·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.939 - 0.342i$
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: \(1\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ -0.939 - 0.342i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 - 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2 + 3.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.5 + 2.59i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 7T + 17T^{2} \)
19 \( 1 + 5T + 19T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.5 - 2.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 11T + 37T^{2} \)
41 \( 1 + (4.5 - 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.5 + 4.33i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.5 - 2.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 3T + 53T^{2} \)
59 \( 1 + (3.5 - 6.06i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.5 - 2.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.5 - 11.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + 7T + 73T^{2} \)
79 \( 1 + (-4.5 - 7.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.5 - 0.866i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 15T + 89T^{2} \)
97 \( 1 + (8.5 + 14.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.789897191943840195728656688343, −8.401828442189882577144039273305, −7.12685167474795544954305853700, −6.09425771595818714518119591384, −5.71560435912110414299267860097, −4.78864368759123323322049562980, −4.06731067756009020546767167620, −2.82680109581761508252724427473, −1.31596483718426853818580891771, 0, 1.97221232381004059382185293125, 2.41390891911840451304352225335, 4.24015753110969675349307242265, 4.74107154704612614656140232953, 6.04788214020341261428847963889, 6.50298383934735771135948085540, 7.08375758753444117725567138885, 8.016936254678466207074755604218, 8.964419060660366002131234864032

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