Properties

Degree $2$
Conductor $1764$
Sign $0.939 + 0.342i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

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: $0.939 + 0.342i$
Motivic weight: \(1\)
Character: $\chi_{1764} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 0.939 + 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.248783424\)
\(L(\frac12)\) \(\approx\) \(2.248783424\)
\(L(\frac{3}{2})\) 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

−9.289373618114066976767908533012, −8.247968132402511121733142599034, −7.62200855230161455202228512280, −7.21508148579139680569296874029, −6.13151671014564510412031210945, −5.30774630498182352102861261643, −3.82924845079674152275958423076, −3.26810381607533865412834763725, −2.43113924164211559633267557438, −0.983440521401001918670678879355, 1.09449135929903545031904738485, 2.56852784842004776780560795607, 3.32918717017753630730606040679, 4.55742412940866144273103127435, 4.89091267378441973098772403982, 5.96383061153251407913677904124, 7.56525048000470160776068205357, 7.69929798365836044307133322510, 8.439381825879307724344873397904, 9.494369520059240105349093439046

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