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Properties

Label 2-42e2-7.4-c1-0-10
Degree $2$
Conductor $1764$
Sign $0.947 - 0.318i$
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  + (1.87 + 3.24i)5-s + (2.64 − 4.58i)11-s + 4.24·13-s + (1.87 − 3.24i)17-s + (1.41 + 2.44i)19-s + (−2.64 − 4.58i)23-s + (−4.5 + 7.79i)25-s + 5.29·29-s + (4.24 − 7.34i)31-s + (−2 − 3.46i)37-s − 3.74·41-s + 8·43-s + (−3.74 − 6.48i)47-s + (−5.29 + 9.16i)53-s + 19.7·55-s + ⋯
L(s)  = 1  + (0.836 + 1.44i)5-s + (0.797 − 1.38i)11-s + 1.17·13-s + (0.453 − 0.785i)17-s + (0.324 + 0.561i)19-s + (−0.551 − 0.955i)23-s + (−0.900 + 1.55i)25-s + 0.982·29-s + (0.762 − 1.31i)31-s + (−0.328 − 0.569i)37-s − 0.584·41-s + 1.21·43-s + (−0.545 − 0.945i)47-s + (−0.726 + 1.25i)53-s + 2.66·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.947 - 0.318i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 0.947 - 0.318i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.319910212\)
\(L(\frac12)\) \(\approx\) \(2.319910212\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-1.87 - 3.24i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.64 + 4.58i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.24T + 13T^{2} \)
17 \( 1 + (-1.87 + 3.24i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.41 - 2.44i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.64 + 4.58i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 5.29T + 29T^{2} \)
31 \( 1 + (-4.24 + 7.34i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2 + 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.74T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + (3.74 + 6.48i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.29 - 9.16i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.74 - 6.48i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.94 - 8.57i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6 - 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 15.8T + 71T^{2} \)
73 \( 1 + (-0.707 + 1.22i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 14.9T + 83T^{2} \)
89 \( 1 + (1.87 + 3.24i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 9.89T + 97T^{2} \)
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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

−9.353461337801603448905747707274, −8.625772104642889122980193641214, −7.72235238522229978103034163833, −6.73706199553481117371916381937, −6.08642587788789594878758480154, −5.74047837634963782427848010473, −4.13807055079567281746143752127, −3.25970798488075805079924964714, −2.53214888657389719368763451348, −1.10796170366097489697318476523, 1.25448002610883002881492636768, 1.76543763495028281873983292284, 3.41400951129522531539868409079, 4.50825968529642343019565023776, 5.04366519712613483521157418963, 6.06269983237647085225030066026, 6.66609430288954048109105079889, 7.897142192905720364863856691806, 8.570251174477102902377326436168, 9.309234812879317262446430958468

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