↑

Properties

Label 2-220-20.3-c1-0-19
Degree $2$
Conductor $220$
Sign $0.865 + 0.500i$
Analytic cond. $1.75670$
Root an. cond. $1.32540$
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  + (1.31 − 0.513i)2-s + (−0.912 + 0.912i)3-s + (1.47 − 1.35i)4-s + (1.46 − 1.68i)5-s + (−0.733 + 1.67i)6-s + (−0.0653 − 0.0653i)7-s + (1.24 − 2.54i)8-s + 1.33i·9-s + (1.06 − 2.97i)10-s − i·11-s + (−0.107 + 2.57i)12-s + (0.473 + 0.473i)13-s + (−0.119 − 0.0524i)14-s + (0.204 + 2.87i)15-s + (0.332 − 3.98i)16-s + (−1.01 + 1.01i)17-s + ⋯
L(s)  = 1  + (0.931 − 0.363i)2-s + (−0.526 + 0.526i)3-s + (0.735 − 0.677i)4-s + (0.655 − 0.755i)5-s + (−0.299 + 0.682i)6-s + (−0.0246 − 0.0246i)7-s + (0.439 − 0.898i)8-s + 0.444i·9-s + (0.335 − 0.941i)10-s − 0.301i·11-s + (−0.0309 + 0.744i)12-s + (0.131 + 0.131i)13-s + (−0.0319 − 0.0140i)14-s + (0.0528 + 0.743i)15-s + (0.0830 − 0.996i)16-s + (−0.245 + 0.245i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 + 0.500i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.865 + 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $0.865 + 0.500i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :1/2),\ 0.865 + 0.500i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83354 - 0.491808i\)
\(L(\frac12)\) \(\approx\) \(1.83354 - 0.491808i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.31 + 0.513i)T \)
5 \( 1 + (-1.46 + 1.68i)T \)
11 \( 1 + iT \)
good3 \( 1 + (0.912 - 0.912i)T - 3iT^{2} \)
7 \( 1 + (0.0653 + 0.0653i)T + 7iT^{2} \)
13 \( 1 + (-0.473 - 0.473i)T + 13iT^{2} \)
17 \( 1 + (1.01 - 1.01i)T - 17iT^{2} \)
19 \( 1 - 2.65T + 19T^{2} \)
23 \( 1 + (5.32 - 5.32i)T - 23iT^{2} \)
29 \( 1 - 5.94iT - 29T^{2} \)
31 \( 1 - 1.44iT - 31T^{2} \)
37 \( 1 + (5.27 - 5.27i)T - 37iT^{2} \)
41 \( 1 + 5.13T + 41T^{2} \)
43 \( 1 + (4.06 - 4.06i)T - 43iT^{2} \)
47 \( 1 + (7.89 + 7.89i)T + 47iT^{2} \)
53 \( 1 + (-2.60 - 2.60i)T + 53iT^{2} \)
59 \( 1 - 7.70T + 59T^{2} \)
61 \( 1 - 7.06T + 61T^{2} \)
67 \( 1 + (-5.47 - 5.47i)T + 67iT^{2} \)
71 \( 1 + 1.80iT - 71T^{2} \)
73 \( 1 + (-5.16 - 5.16i)T + 73iT^{2} \)
79 \( 1 - 14.2T + 79T^{2} \)
83 \( 1 + (-11.9 + 11.9i)T - 83iT^{2} \)
89 \( 1 + 5.69iT - 89T^{2} \)
97 \( 1 + (0.398 - 0.398i)T - 97iT^{2} \)
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

−12.11958497397502665443461383386, −11.42543456784648680343337957978, −10.33819835365175548490084088577, −9.723847458896414653144354802389, −8.303915177972996513214522828321, −6.70404330228481278574375419233, −5.49962230331611239823887521395, −5.00109772773873541451647611080, −3.65296834201862172864947065186, −1.79140148749787274270072734663, 2.22978421346614828705390282665, 3.70621409831943103843164909669, 5.28531815795923366984717247156, 6.29121920684143919758051267333, 6.84482919533330499286492839889, 7.989354990795769072340058121419, 9.572814531805546894399298231278, 10.72596727433304238946470118086, 11.68690842864470167210631757891, 12.40726944964209372888768418275

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