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Properties

Label 2-42e2-63.41-c1-0-4
Degree $2$
Conductor $1764$
Sign $-0.139 - 0.990i$
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  + (−1.56 − 0.746i)3-s + (0.842 − 1.45i)5-s + (1.88 + 2.33i)9-s + (−3.38 + 1.95i)11-s + (5.24 + 3.02i)13-s + (−2.40 + 1.65i)15-s − 0.402·17-s − 0.168i·19-s + (−7.69 − 4.44i)23-s + (1.07 + 1.86i)25-s + (−1.20 − 5.05i)27-s + (−6.15 + 3.55i)29-s + (−5.44 − 3.14i)31-s + (6.74 − 0.525i)33-s − 6.26·37-s + ⋯
L(s)  = 1  + (−0.902 − 0.431i)3-s + (0.376 − 0.652i)5-s + (0.628 + 0.778i)9-s + (−1.01 + 0.588i)11-s + (1.45 + 0.839i)13-s + (−0.621 + 0.426i)15-s − 0.0976·17-s − 0.0385i·19-s + (−1.60 − 0.926i)23-s + (0.215 + 0.373i)25-s + (−0.230 − 0.972i)27-s + (−1.14 + 0.659i)29-s + (−0.977 − 0.564i)31-s + (1.17 − 0.0913i)33-s − 1.02·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.139 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.139 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.139 - 0.990i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ -0.139 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5847496569\)
\(L(\frac12)\) \(\approx\) \(0.5847496569\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.56 + 0.746i)T \)
7 \( 1 \)
good5 \( 1 + (-0.842 + 1.45i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.38 - 1.95i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-5.24 - 3.02i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 0.402T + 17T^{2} \)
19 \( 1 + 0.168iT - 19T^{2} \)
23 \( 1 + (7.69 + 4.44i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (6.15 - 3.55i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.44 + 3.14i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 6.26T + 37T^{2} \)
41 \( 1 + (1.64 - 2.85i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.80 - 3.12i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.38 - 7.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 5.71iT - 53T^{2} \)
59 \( 1 + (-2.25 + 3.89i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.43 - 2.56i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.95 + 5.11i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 11.4iT - 71T^{2} \)
73 \( 1 - 6.99iT - 73T^{2} \)
79 \( 1 + (0.603 + 1.04i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.181 + 0.314i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 2.77T + 89T^{2} \)
97 \( 1 + (-0.508 + 0.293i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.513769053738589601788038837930, −8.719065218304145376795696318804, −7.86535037515871522211910498126, −7.07958847538085120645929697908, −6.14829377286220336089982846430, −5.58936384275703769644907796702, −4.72664061944152547350451080646, −3.87984080483920390756244554746, −2.18800663183378830740874382766, −1.36320412147329382305526678779, 0.24818145278534970541957540140, 1.88176204295001027192528452724, 3.32522407753686649790336347278, 3.91576742243472489532784821206, 5.40785595974211491049284878211, 5.68168139779484294659644434784, 6.46053608091570583754817275334, 7.43525341924736947118971213510, 8.290199366196608885530001488438, 9.180444055983000122733650508153

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