↑

Properties

Label 2-38e2-76.63-c0-0-1
Degree $2$
Conductor $1444$
Sign $0.801 + 0.597i$
Analytic cond. $0.720649$
Root an. cond. $0.848910$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.766 + 0.642i)2-s + (0.173 − 0.984i)4-s + (−0.280 − 1.59i)5-s + (0.500 + 0.866i)8-s + (0.766 + 0.642i)9-s + (1.23 + 1.04i)10-s + (0.580 − 0.211i)13-s + (−0.939 − 0.342i)16-s + (0.473 − 0.397i)17-s − 18-s − 1.61·20-s + (−1.52 + 0.553i)25-s + (−0.309 + 0.535i)26-s + (−0.473 − 0.397i)29-s + (0.939 − 0.342i)32-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)2-s + (0.173 − 0.984i)4-s + (−0.280 − 1.59i)5-s + (0.500 + 0.866i)8-s + (0.766 + 0.642i)9-s + (1.23 + 1.04i)10-s + (0.580 − 0.211i)13-s + (−0.939 − 0.342i)16-s + (0.473 − 0.397i)17-s − 18-s − 1.61·20-s + (−1.52 + 0.553i)25-s + (−0.309 + 0.535i)26-s + (−0.473 − 0.397i)29-s + (0.939 − 0.342i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 + 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 + 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1444\)    =    \(2^{2} \cdot 19^{2}\)
Sign: $0.801 + 0.597i$
Analytic conductor: \(0.720649\)
Root analytic conductor: \(0.848910\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1444} (595, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1444,\ (\ :0),\ 0.801 + 0.597i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7614743099\)
\(L(\frac12)\) \(\approx\) \(0.7614743099\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.766 - 0.642i)T \)
19 \( 1 \)
good3 \( 1 + (-0.766 - 0.642i)T^{2} \)
5 \( 1 + (0.280 + 1.59i)T + (-0.939 + 0.342i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.580 + 0.211i)T + (0.766 - 0.642i)T^{2} \)
17 \( 1 + (-0.473 + 0.397i)T + (0.173 - 0.984i)T^{2} \)
23 \( 1 + (0.939 + 0.342i)T^{2} \)
29 \( 1 + (0.473 + 0.397i)T + (0.173 + 0.984i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - 1.61T + T^{2} \)
41 \( 1 + (1.52 + 0.553i)T + (0.766 + 0.642i)T^{2} \)
43 \( 1 + (0.939 - 0.342i)T^{2} \)
47 \( 1 + (-0.173 - 0.984i)T^{2} \)
53 \( 1 + (-0.280 + 1.59i)T + (-0.939 - 0.342i)T^{2} \)
59 \( 1 + (-0.173 + 0.984i)T^{2} \)
61 \( 1 + (-0.107 + 0.608i)T + (-0.939 - 0.342i)T^{2} \)
67 \( 1 + (-0.173 - 0.984i)T^{2} \)
71 \( 1 + (0.939 - 0.342i)T^{2} \)
73 \( 1 + (-1.52 - 0.553i)T + (0.766 + 0.642i)T^{2} \)
79 \( 1 + (-0.766 - 0.642i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (1.52 - 0.553i)T + (0.766 - 0.642i)T^{2} \)
97 \( 1 + (0.473 - 0.397i)T + (0.173 - 0.984i)T^{2} \)
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

−9.589196380418171138687093339885, −8.649982208160732320064676592225, −8.142090695641408242371078015642, −7.49678161397321880275168264672, −6.48232680570105308973536996848, −5.36791868070987748380259866184, −4.93436421257032907444698566167, −3.90299949164734803468449818062, −1.97397212393113611542562201465, −0.915433016892156785637691307476, 1.48008503117779649418788219783, 2.79088688570647037966452049275, 3.52607964610249815229108878008, 4.28673151630723036090616400514, 6.08752356484975135510317400569, 6.77000062238611348221767041858, 7.44006570502743885184519703207, 8.160506381989458275847102706197, 9.234528617189060214780303107775, 9.910574134290348824062945989396

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