↑

Properties

Label 2-42e2-28.27-c1-0-4
Degree $2$
Conductor $1764$
Sign $-0.679 + 0.733i$
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  + (0.0777 + 1.41i)2-s + (−1.98 + 0.219i)4-s + 0.438i·5-s + (−0.464 − 2.79i)8-s + (−0.619 + 0.0341i)10-s + 2.11i·11-s − 3.84i·13-s + (3.90 − 0.872i)16-s + 5.64i·17-s − 2.97·19-s + (−0.0963 − 0.872i)20-s + (−2.98 + 0.164i)22-s + 4.77i·23-s + 4.80·25-s + (5.43 − 0.299i)26-s + ⋯
L(s)  = 1  + (0.0549 + 0.998i)2-s + (−0.993 + 0.109i)4-s + 0.196i·5-s + (−0.164 − 0.986i)8-s + (−0.196 + 0.0107i)10-s + 0.637i·11-s − 1.06i·13-s + (0.975 − 0.218i)16-s + 1.36i·17-s − 0.682·19-s + (−0.0215 − 0.195i)20-s + (−0.637 + 0.0350i)22-s + 0.994i·23-s + 0.961·25-s + (1.06 − 0.0586i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.679 + 0.733i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.679 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.679 + 0.733i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ -0.679 + 0.733i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4619988317\)
\(L(\frac12)\) \(\approx\) \(0.4619988317\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0777 - 1.41i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 0.438iT - 5T^{2} \)
11 \( 1 - 2.11iT - 11T^{2} \)
13 \( 1 + 3.84iT - 13T^{2} \)
17 \( 1 - 5.64iT - 17T^{2} \)
19 \( 1 + 2.97T + 19T^{2} \)
23 \( 1 - 4.77iT - 23T^{2} \)
29 \( 1 + 7.02T + 29T^{2} \)
31 \( 1 + 7.42T + 31T^{2} \)
37 \( 1 + 5.28T + 37T^{2} \)
41 \( 1 + 6.81iT - 41T^{2} \)
43 \( 1 - 4.38iT - 43T^{2} \)
47 \( 1 - 1.68T + 47T^{2} \)
53 \( 1 + 10.7T + 53T^{2} \)
59 \( 1 + 8.11T + 59T^{2} \)
61 \( 1 + 6.18iT - 61T^{2} \)
67 \( 1 - 7.85iT - 67T^{2} \)
71 \( 1 - 1.16iT - 71T^{2} \)
73 \( 1 + 10.0iT - 73T^{2} \)
79 \( 1 - 15.5iT - 79T^{2} \)
83 \( 1 - 5.49T + 83T^{2} \)
89 \( 1 + 10.4iT - 89T^{2} \)
97 \( 1 - 2.22iT - 97T^{2} \)
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

−9.575845855103511678094855727796, −8.896683375638672960862050028973, −8.031938969461209635667755379271, −7.45435943264532877501184242605, −6.64819198476270151144249467147, −5.77465886869267029903028012860, −5.16079580537431330549256847188, −4.05575524633514783781943752900, −3.30876120807947010944703538173, −1.68575329317485246365903913710, 0.16881553361392887664000249791, 1.60129055609798125631774415925, 2.65127594981485515581615142920, 3.62296882261664716608513027727, 4.55999159987294663139580730825, 5.24167855536837943984308559875, 6.31292283988247300861086769156, 7.26914011689683342834163744644, 8.310365895425665338889449479074, 9.149000455554762374086287916236

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