Properties

Label 2-644-161.10-c1-0-7
Degree $2$
Conductor $644$
Sign $0.488 - 0.872i$
Analytic cond. $5.14236$
Root an. cond. $2.26767$
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  + (1.20 + 2.32i)3-s + (1.77 − 0.169i)5-s + (2.18 − 1.49i)7-s + (−2.23 + 3.14i)9-s + (−0.541 + 0.187i)11-s + (−0.0682 − 0.232i)13-s + (2.52 + 3.93i)15-s + (4.98 + 1.99i)17-s + (3.52 − 1.41i)19-s + (6.10 + 3.27i)21-s + (−4.68 − 1.03i)23-s + (−1.78 + 0.343i)25-s + (−2.22 − 0.319i)27-s + (−0.349 − 2.43i)29-s + (−7.53 + 0.359i)31-s + ⋯
L(s)  = 1  + (0.692 + 1.34i)3-s + (0.794 − 0.0758i)5-s + (0.823 − 0.566i)7-s + (−0.745 + 1.04i)9-s + (−0.163 + 0.0565i)11-s + (−0.0189 − 0.0644i)13-s + (0.652 + 1.01i)15-s + (1.20 + 0.483i)17-s + (0.809 − 0.323i)19-s + (1.33 + 0.714i)21-s + (−0.976 − 0.215i)23-s + (−0.356 + 0.0686i)25-s + (−0.428 − 0.0615i)27-s + (−0.0649 − 0.451i)29-s + (−1.35 + 0.0644i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.488 - 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.488 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(644\)    =    \(2^{2} \cdot 7 \cdot 23\)
Sign: $0.488 - 0.872i$
Analytic conductor: \(5.14236\)
Root analytic conductor: \(2.26767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{644} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 644,\ (\ :1/2),\ 0.488 - 0.872i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.97194 + 1.15660i\)
\(L(\frac12)\) \(\approx\) \(1.97194 + 1.15660i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-2.18 + 1.49i)T \)
23 \( 1 + (4.68 + 1.03i)T \)
good3 \( 1 + (-1.20 - 2.32i)T + (-1.74 + 2.44i)T^{2} \)
5 \( 1 + (-1.77 + 0.169i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (0.541 - 0.187i)T + (8.64 - 6.79i)T^{2} \)
13 \( 1 + (0.0682 + 0.232i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (-4.98 - 1.99i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (-3.52 + 1.41i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (0.349 + 2.43i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (7.53 - 0.359i)T + (30.8 - 2.94i)T^{2} \)
37 \( 1 + (5.14 + 3.66i)T + (12.1 + 34.9i)T^{2} \)
41 \( 1 + (-4.49 - 2.05i)T + (26.8 + 30.9i)T^{2} \)
43 \( 1 + (2.66 - 4.14i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (7.64 - 4.41i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.35 - 4.56i)T + (-2.52 + 52.9i)T^{2} \)
59 \( 1 + (2.51 + 0.610i)T + (52.4 + 27.0i)T^{2} \)
61 \( 1 + (6.01 + 3.09i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (-0.914 - 4.74i)T + (-62.2 + 24.9i)T^{2} \)
71 \( 1 + (5.21 + 6.02i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-7.70 + 9.80i)T + (-17.2 - 70.9i)T^{2} \)
79 \( 1 + (4.43 - 4.64i)T + (-3.75 - 78.9i)T^{2} \)
83 \( 1 + (1.10 + 2.41i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (-0.277 + 5.82i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (3.97 - 8.70i)T + (-63.5 - 73.3i)T^{2} \)
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

−10.44929561082590806833763728147, −9.837317559519071638567647082273, −9.236100527389700973021200858045, −8.188993085811699264564009928823, −7.49183388008741886538368437422, −5.89567363012041944072034065579, −5.09047245125169499892852142516, −4.13670373791129952650684553372, −3.20610546582424305301186318384, −1.75731406552405092190634329462, 1.46295000246518289492853782614, 2.20841568404361121377745960093, 3.40398076619499563218597659173, 5.25074692342959391196147973802, 5.88778936498481080857914525631, 7.09170157801099575873048745163, 7.77532705774948240008496404368, 8.487175658215348353104589519544, 9.420412412117112962168307895576, 10.27858382411162701966049287758

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