↑

Properties

Label 2-42e2-63.59-c1-0-3
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

Related objects

Downloads

Learn more

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: $\chi_{1764} (1697, \cdot )$
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 \( 1 + (7.48 - 4.32i)T + (48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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