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Properties

Label 2-42e2-63.20-c1-0-33
Degree $2$
Conductor $1764$
Sign $-0.107 + 0.994i$
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  + (1.55 − 0.771i)3-s + (−1.43 − 2.48i)5-s + (1.80 − 2.39i)9-s + (2.34 + 1.35i)11-s + (3.18 − 1.84i)13-s + (−4.14 − 2.74i)15-s + 6.44·17-s − 3.16i·19-s + (−2.59 + 1.49i)23-s + (−1.61 + 2.79i)25-s + (0.956 − 5.10i)27-s + (−2.48 − 1.43i)29-s + (−8.26 + 4.77i)31-s + (4.68 + 0.289i)33-s + 3.41·37-s + ⋯
L(s)  = 1  + (0.895 − 0.445i)3-s + (−0.641 − 1.11i)5-s + (0.602 − 0.797i)9-s + (0.708 + 0.408i)11-s + (0.884 − 0.510i)13-s + (−1.06 − 0.708i)15-s + 1.56·17-s − 0.725i·19-s + (−0.540 + 0.311i)23-s + (−0.322 + 0.558i)25-s + (0.184 − 0.982i)27-s + (−0.461 − 0.266i)29-s + (−1.48 + 0.857i)31-s + (0.816 + 0.0504i)33-s + 0.561·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.107 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.107 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.107 + 0.994i$
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: \((2,\ 1764,\ (\ :1/2),\ -0.107 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.305833755\)
\(L(\frac12)\) \(\approx\) \(2.305833755\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.55 + 0.771i)T \)
7 \( 1 \)
good5 \( 1 + (1.43 + 2.48i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.34 - 1.35i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.18 + 1.84i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 6.44T + 17T^{2} \)
19 \( 1 + 3.16iT - 19T^{2} \)
23 \( 1 + (2.59 - 1.49i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.48 + 1.43i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (8.26 - 4.77i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.41T + 37T^{2} \)
41 \( 1 + (-0.794 - 1.37i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.67 - 8.10i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.65 + 9.79i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 2.49iT - 53T^{2} \)
59 \( 1 + (4.33 + 7.51i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.566 - 0.327i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.86 + 6.68i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 7.86iT - 71T^{2} \)
73 \( 1 - 12.7iT - 73T^{2} \)
79 \( 1 + (2.59 - 4.49i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.92 + 13.7i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 6.29T + 89T^{2} \)
97 \( 1 + (-13.2 - 7.62i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.058950345315446804323371897870, −8.197169906475063937253993225532, −7.78729394642568405978884662609, −6.90629234227696132447236384477, −5.86216690443557470495802933341, −4.87344835299036023569903518374, −3.85323107569433482020567164770, −3.30078784559683141881660013010, −1.75334161407476909997222436513, −0.854027405594720923317485600640, 1.56008832104693517386352056374, 2.85306664047268989969880333685, 3.77046328175684716409734657787, 3.95247722308398158852979539096, 5.54109078894192127555125970916, 6.37121708411934219544151664123, 7.45611562822903491357477198734, 7.77639795729766916959301822579, 8.787058115036111010595414493461, 9.404138385842724698742597732654

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