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Properties

Label 2-42e2-63.47-c1-0-1
Degree $2$
Conductor $1764$
Sign $-0.0364 - 0.999i$
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  + (−0.127 − 1.72i)3-s − 2.18·5-s + (−2.96 + 0.441i)9-s − 1.46i·11-s + (2.92 − 1.69i)13-s + (0.278 + 3.77i)15-s + (1.32 + 2.28i)17-s + (−6.87 − 3.97i)19-s + 4.00i·23-s − 0.234·25-s + (1.14 + 5.06i)27-s + (−6.71 − 3.87i)29-s + (−0.612 − 0.353i)31-s + (−2.53 + 0.187i)33-s + (1.41 − 2.45i)37-s + ⋯
L(s)  = 1  + (−0.0737 − 0.997i)3-s − 0.976·5-s + (−0.989 + 0.147i)9-s − 0.441i·11-s + (0.811 − 0.468i)13-s + (0.0720 + 0.973i)15-s + (0.320 + 0.555i)17-s + (−1.57 − 0.911i)19-s + 0.836i·23-s − 0.0468·25-s + (0.219 + 0.975i)27-s + (−1.24 − 0.719i)29-s + (−0.109 − 0.0634i)31-s + (−0.440 + 0.0325i)33-s + (0.233 − 0.403i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0364 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0364 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.0364 - 0.999i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1685, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ -0.0364 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2459467860\)
\(L(\frac12)\) \(\approx\) \(0.2459467860\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.127 + 1.72i)T \)
7 \( 1 \)
good5 \( 1 + 2.18T + 5T^{2} \)
11 \( 1 + 1.46iT - 11T^{2} \)
13 \( 1 + (-2.92 + 1.69i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.32 - 2.28i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.87 + 3.97i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.00iT - 23T^{2} \)
29 \( 1 + (6.71 + 3.87i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.612 + 0.353i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.41 + 2.45i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.74 - 6.48i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.27 - 2.20i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.27 - 10.8i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.41 + 1.39i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.71 - 11.6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.75 - 3.89i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.92 - 5.05i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 11.6iT - 71T^{2} \)
73 \( 1 + (-3.95 + 2.28i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.69 + 8.12i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.70 + 2.95i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.61 + 8.00i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.38 - 3.68i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.183335539028910553044067266331, −8.549170563025943496626128396451, −7.78380062274330302577407999755, −7.38349323058385881210708021600, −6.16283291919874198626869077682, −5.84430966334968288371554352627, −4.42019665825818386001808048812, −3.57626956489264441837015891750, −2.52976405023945584886041639022, −1.20444816981754249607008401775, 0.10003631198563678162197473246, 2.08307476559541118867306426961, 3.51702495011045925722168782766, 3.98918212287932707249575938662, 4.78435164210194950666298580846, 5.78620423958535231367355621090, 6.65519956153919934001634952861, 7.66404315645926505623797528245, 8.453586323956139398127457449530, 9.026908625347074943484969601324

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