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Properties

Label 2-42e2-63.47-c1-0-10
Degree $2$
Conductor $1764$
Sign $0.999 + 0.00293i$
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  + (−1.69 − 0.357i)3-s − 2.42·5-s + (2.74 + 1.21i)9-s − 2.42i·11-s + (−4.73 + 2.73i)13-s + (4.10 + 0.866i)15-s + (−1.29 − 2.23i)17-s + (0.348 + 0.201i)19-s + 3.54i·23-s + 0.880·25-s + (−4.21 − 3.03i)27-s + (−6.31 − 3.64i)29-s + (3.63 + 2.10i)31-s + (−0.864 + 4.10i)33-s + (1.59 − 2.76i)37-s + ⋯
L(s)  = 1  + (−0.978 − 0.206i)3-s − 1.08·5-s + (0.914 + 0.403i)9-s − 0.730i·11-s + (−1.31 + 0.758i)13-s + (1.06 + 0.223i)15-s + (−0.312 − 0.542i)17-s + (0.0800 + 0.0461i)19-s + 0.738i·23-s + 0.176·25-s + (−0.812 − 0.583i)27-s + (−1.17 − 0.677i)29-s + (0.653 + 0.377i)31-s + (−0.150 + 0.714i)33-s + (0.262 − 0.454i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00293i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00293i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.999 + 0.00293i$
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: \((2,\ 1764,\ (\ :1/2),\ 0.999 + 0.00293i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6386586856\)
\(L(\frac12)\) \(\approx\) \(0.6386586856\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.69 + 0.357i)T \)
7 \( 1 \)
good5 \( 1 + 2.42T + 5T^{2} \)
11 \( 1 + 2.42iT - 11T^{2} \)
13 \( 1 + (4.73 - 2.73i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.29 + 2.23i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.348 - 0.201i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.54iT - 23T^{2} \)
29 \( 1 + (6.31 + 3.64i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.63 - 2.10i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.59 + 2.76i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.03 - 6.99i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.22 - 7.31i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.25 + 3.91i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-12.1 + 7.01i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.0779 - 0.134i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-10.2 + 5.90i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.53 + 4.39i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.73iT - 71T^{2} \)
73 \( 1 + (7.62 - 4.40i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.66 - 9.81i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.50 + 13.0i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.83 + 13.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.97 - 2.87i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.468647172255697348521821423398, −8.318834956497209460783458288124, −7.53953328342092030324577418095, −7.02938939027530327428894408340, −6.11198335556514445031365123046, −5.15342421941976614723154340746, −4.44905062843586368383756173425, −3.55132987467568127584191586859, −2.16362870499009616234400164315, −0.59032168973052014859751601801, 0.52398223206591902019804255344, 2.22927921605963506852372017890, 3.64208485066479281229021804285, 4.42093050603838790903862225708, 5.11271810639736982956316085025, 6.01224853189375007445442445746, 7.16437767050378434268962309961, 7.41084389033027097745434795913, 8.427253570219109601258354037831, 9.450752205687625468752410946005

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