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Properties

Label 2-42e2-63.47-c1-0-38
Degree $2$
Conductor $1764$
Sign $-0.266 + 0.963i$
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  + (0.744 + 1.56i)3-s + 0.553·5-s + (−1.89 + 2.32i)9-s − 4.65i·11-s + (−3.58 + 2.06i)13-s + (0.412 + 0.866i)15-s + (−3.62 − 6.27i)17-s + (−5.81 − 3.35i)19-s − 5.60i·23-s − 4.69·25-s + (−5.05 − 1.22i)27-s + (1.16 + 0.673i)29-s + (−0.830 − 0.479i)31-s + (7.28 − 3.47i)33-s + (3.53 − 6.12i)37-s + ⋯
L(s)  = 1  + (0.430 + 0.902i)3-s + 0.247·5-s + (−0.630 + 0.776i)9-s − 1.40i·11-s + (−0.993 + 0.573i)13-s + (0.106 + 0.223i)15-s + (−0.878 − 1.52i)17-s + (−1.33 − 0.770i)19-s − 1.16i·23-s − 0.938·25-s + (−0.972 − 0.234i)27-s + (0.216 + 0.125i)29-s + (−0.149 − 0.0861i)31-s + (1.26 − 0.604i)33-s + (0.581 − 1.00i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.266 + 0.963i$
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: \((2,\ 1764,\ (\ :1/2),\ -0.266 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6649538766\)
\(L(\frac12)\) \(\approx\) \(0.6649538766\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.744 - 1.56i)T \)
7 \( 1 \)
good5 \( 1 - 0.553T + 5T^{2} \)
11 \( 1 + 4.65iT - 11T^{2} \)
13 \( 1 + (3.58 - 2.06i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.62 + 6.27i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.81 + 3.35i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 5.60iT - 23T^{2} \)
29 \( 1 + (-1.16 - 0.673i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.830 + 0.479i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.53 + 6.12i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.39 - 4.14i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.02 - 1.78i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.90 - 8.49i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.30 - 4.21i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.89 - 6.75i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.37 - 3.10i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.68 + 2.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.407iT - 71T^{2} \)
73 \( 1 + (-7.47 + 4.31i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.318 + 0.551i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.78 - 4.82i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.46 - 6.00i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.48 + 4.32i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.231199815002792477885615045915, −8.495037694786262868642103882735, −7.62764151227771498925543702868, −6.58748352763764079796563503702, −5.78569676415496940389428669820, −4.68425668627073547214074353838, −4.27587751835185914337260315487, −2.87823468242832820709772617128, −2.37058685301488940392244695729, −0.20815553163180755221691102674, 1.80263247179879798009296174856, 2.18597100066109821771850863662, 3.59159230927471106882802045630, 4.51724843682111393107213823927, 5.68965094367804645793798899122, 6.43732260265795613279632447145, 7.21173927086363741735890342614, 7.939401578130154630287582864396, 8.542478441750735738314216060550, 9.593700564124261334680342645162

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