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Properties

Label 2-42e2-63.16-c1-0-22
Degree $2$
Conductor $1764$
Sign $0.951 - 0.308i$
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  + (−0.796 + 1.53i)3-s + 2.46·5-s + (−1.73 − 2.45i)9-s + 4.64·11-s + (−3.55 − 6.15i)13-s + (−1.96 + 3.78i)15-s + (2.25 + 3.90i)17-s + (−2.16 + 3.74i)19-s + 5.86·23-s + 1.05·25-s + (5.14 − 0.708i)27-s + (3.48 − 6.04i)29-s + (3.69 − 6.39i)31-s + (−3.70 + 7.14i)33-s + (0.363 − 0.629i)37-s + ⋯
L(s)  = 1  + (−0.460 + 0.887i)3-s + 1.10·5-s + (−0.576 − 0.816i)9-s + 1.40·11-s + (−0.985 − 1.70i)13-s + (−0.506 + 0.977i)15-s + (0.547 + 0.948i)17-s + (−0.496 + 0.859i)19-s + 1.22·23-s + 0.210·25-s + (0.990 − 0.136i)27-s + (0.647 − 1.12i)29-s + (0.662 − 1.14i)31-s + (−0.644 + 1.24i)33-s + (0.0597 − 0.103i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 - 0.308i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.951 - 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.951 - 0.308i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 0.951 - 0.308i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.896662220\)
\(L(\frac12)\) \(\approx\) \(1.896662220\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.796 - 1.53i)T \)
7 \( 1 \)
good5 \( 1 - 2.46T + 5T^{2} \)
11 \( 1 - 4.64T + 11T^{2} \)
13 \( 1 + (3.55 + 6.15i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.25 - 3.90i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.16 - 3.74i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 5.86T + 23T^{2} \)
29 \( 1 + (-3.48 + 6.04i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.69 + 6.39i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.363 + 0.629i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.136 + 0.236i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.41 + 4.18i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.83 + 3.18i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.52 + 4.37i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.56 - 7.90i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.90 - 11.9i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.663 + 1.14i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 13.5T + 71T^{2} \)
73 \( 1 + (-2.16 - 3.74i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.21 + 5.57i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.742 - 1.28i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.91 - 8.51i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.246 + 0.426i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.630395618015735796191461599983, −8.726633796058602598052932385431, −7.892273922701232528557570672321, −6.64690824872769438380420068111, −5.90379679988909563599271871989, −5.47277821121864085254714046135, −4.39841143714459582083196683004, −3.52223094277898337708427410456, −2.40920223524449597790810172069, −0.929700252686879874376443066466, 1.13462845441402703929232857476, 1.99040980158979522600478008067, 2.99147052470077674715453490391, 4.66018553147435605008230952404, 5.12636011167344980939851399504, 6.47899185046256410484290977577, 6.62618436427311099739330191183, 7.33265442574307106139427836673, 8.656550697711590159094292285027, 9.288896089999997574530627149811

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