Properties

Label 2-42e2-7.3-c2-0-7
Degree $2$
Conductor $1764$
Sign $-0.379 - 0.925i$
Analytic cond. $48.0655$
Root an. cond. $6.93293$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−7.33 + 4.23i)5-s + (1.94 − 3.37i)11-s − 19.1i·13-s + (11.5 + 6.69i)17-s + (−5.80 + 3.35i)19-s + (13 + 22.5i)23-s + (23.3 − 40.5i)25-s + 11.7·29-s + (−31.6 − 18.2i)31-s + (16 + 27.7i)37-s − 20.9i·41-s + 79.2·43-s + (−12.3 + 7.15i)47-s + (−6.89 + 11.9i)53-s + 33.0i·55-s + ⋯
L(s)  = 1  + (−1.46 + 0.847i)5-s + (0.177 − 0.307i)11-s − 1.47i·13-s + (0.681 + 0.393i)17-s + (−0.305 + 0.176i)19-s + (0.565 + 0.978i)23-s + (0.935 − 1.62i)25-s + 0.406·29-s + (−1.02 − 0.590i)31-s + (0.432 + 0.748i)37-s − 0.511i·41-s + 1.84·43-s + (−0.263 + 0.152i)47-s + (−0.130 + 0.225i)53-s + 0.600i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.379 - 0.925i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.379 - 0.925i$
Analytic conductor: \(48.0655\)
Root analytic conductor: \(6.93293\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1),\ -0.379 - 0.925i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8502778052\)
\(L(\frac12)\) \(\approx\) \(0.8502778052\)
\(L(2)\) 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 \( 1 + (7.33 - 4.23i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-1.94 + 3.37i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 19.1iT - 169T^{2} \)
17 \( 1 + (-11.5 - 6.69i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (5.80 - 3.35i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-13 - 22.5i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 11.7T + 841T^{2} \)
31 \( 1 + (31.6 + 18.2i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-16 - 27.7i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 20.9iT - 1.68e3T^{2} \)
43 \( 1 - 79.2T + 1.84e3T^{2} \)
47 \( 1 + (12.3 - 7.15i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (6.89 - 11.9i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (7.31 + 4.22i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (27.4 - 15.8i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-15.6 + 27.1i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 95.5T + 5.04e3T^{2} \)
73 \( 1 + (-15.8 - 9.15i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (39.8 + 69.1i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 141. iT - 6.88e3T^{2} \)
89 \( 1 + (100. - 58.2i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 137. iT - 9.40e3T^{2} \)
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

−9.332618703836085768956788766960, −8.293211137411148984367343669338, −7.74773568213746183727228612757, −7.24524706491516779315452867267, −6.16892827007420163920575762288, −5.36787872367470488219501130041, −4.13585580017972491229816951047, −3.45037997143592350313918222040, −2.75230648052172626892430344496, −0.961928121107703897143441623122, 0.29023280984017384568061675354, 1.48940946887470746759471129796, 2.93592081592963131672866423087, 4.18854697685026741215060694227, 4.40089918707932358495974361520, 5.45509365109369793024876072505, 6.70871424508607367867325581220, 7.32873498499433253139159980887, 8.085905311036559018344971507461, 8.955484567157584201378441053904

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