↑

Properties

Label 2-220-20.7-c1-0-14
Degree $2$
Conductor $220$
Sign $0.321 - 0.947i$
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  + (1.23 + 0.682i)2-s + (1.16 + 1.16i)3-s + (1.06 + 1.68i)4-s + (−2.21 + 0.280i)5-s + (0.649 + 2.24i)6-s + (2.18 − 2.18i)7-s + (0.172 + 2.82i)8-s − 0.277i·9-s + (−2.93 − 1.16i)10-s − i·11-s + (−0.723 + 3.21i)12-s + (−2.68 + 2.68i)13-s + (4.18 − 1.21i)14-s + (−2.91 − 2.26i)15-s + (−1.71 + 3.61i)16-s + (−1.20 − 1.20i)17-s + ⋯
L(s)  = 1  + (0.876 + 0.482i)2-s + (0.673 + 0.673i)3-s + (0.534 + 0.844i)4-s + (−0.992 + 0.125i)5-s + (0.265 + 0.914i)6-s + (0.824 − 0.824i)7-s + (0.0610 + 0.998i)8-s − 0.0924i·9-s + (−0.929 − 0.368i)10-s − 0.301i·11-s + (−0.208 + 0.929i)12-s + (−0.745 + 0.745i)13-s + (1.11 − 0.324i)14-s + (−0.752 − 0.583i)15-s + (−0.427 + 0.903i)16-s + (−0.292 − 0.292i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.321 - 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.321 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $0.321 - 0.947i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :1/2),\ 0.321 - 0.947i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.72547 + 1.23681i\)
\(L(\frac12)\) \(\approx\) \(1.72547 + 1.23681i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.23 - 0.682i)T \)
5 \( 1 + (2.21 - 0.280i)T \)
11 \( 1 + iT \)
good3 \( 1 + (-1.16 - 1.16i)T + 3iT^{2} \)
7 \( 1 + (-2.18 + 2.18i)T - 7iT^{2} \)
13 \( 1 + (2.68 - 2.68i)T - 13iT^{2} \)
17 \( 1 + (1.20 + 1.20i)T + 17iT^{2} \)
19 \( 1 - 0.758T + 19T^{2} \)
23 \( 1 + (4.06 + 4.06i)T + 23iT^{2} \)
29 \( 1 + 3.39iT - 29T^{2} \)
31 \( 1 + 4.55iT - 31T^{2} \)
37 \( 1 + (-4.42 - 4.42i)T + 37iT^{2} \)
41 \( 1 - 1.63T + 41T^{2} \)
43 \( 1 + (-7.31 - 7.31i)T + 43iT^{2} \)
47 \( 1 + (1.24 - 1.24i)T - 47iT^{2} \)
53 \( 1 + (4.89 - 4.89i)T - 53iT^{2} \)
59 \( 1 + 8.66T + 59T^{2} \)
61 \( 1 + 13.4T + 61T^{2} \)
67 \( 1 + (-10.3 + 10.3i)T - 67iT^{2} \)
71 \( 1 - 0.631iT - 71T^{2} \)
73 \( 1 + (11.0 - 11.0i)T - 73iT^{2} \)
79 \( 1 + 9.62T + 79T^{2} \)
83 \( 1 + (-4.44 - 4.44i)T + 83iT^{2} \)
89 \( 1 - 6.02iT - 89T^{2} \)
97 \( 1 + (1.24 + 1.24i)T + 97iT^{2} \)
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

−12.45597682988815867395551828428, −11.60142811270599466463207280708, −10.82783327272404292809593572631, −9.416253796777751984029712880556, −8.166073742704267102526673186592, −7.58768300052529649837752582690, −6.38520159324684617276495880516, −4.44637461820754411506830024386, −4.27939123671738021527583071921, −2.84349855274072701476169157303, 1.86347670643013342259963302416, 3.08362205554137600776640548247, 4.55917655560923689657806459914, 5.56214843689261253523277643300, 7.22916211218784803523804399086, 7.913699485798925077394548497853, 9.011149713307147463122804768737, 10.49338528944013501659930069706, 11.44598149318954798607452722178, 12.31405577996967752973389479414

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