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Properties

Label 2-220-20.19-c2-0-11
Degree $2$
Conductor $220$
Sign $-0.973 + 0.230i$
Analytic cond. $5.99456$
Root an. cond. $2.44838$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.698 + 1.87i)2-s + 0.798·3-s + (−3.02 + 2.61i)4-s + (−2.31 + 4.43i)5-s + (0.557 + 1.49i)6-s − 0.222·7-s + (−7.01 − 3.84i)8-s − 8.36·9-s + (−9.92 − 1.23i)10-s + 3.31i·11-s + (−2.41 + 2.08i)12-s − 4.74i·13-s + (−0.155 − 0.416i)14-s + (−1.84 + 3.53i)15-s + (2.30 − 15.8i)16-s − 4.87i·17-s + ⋯
L(s)  = 1  + (0.349 + 0.937i)2-s + 0.266·3-s + (−0.756 + 0.654i)4-s + (−0.462 + 0.886i)5-s + (0.0929 + 0.249i)6-s − 0.0317·7-s + (−0.877 − 0.480i)8-s − 0.929·9-s + (−0.992 − 0.123i)10-s + 0.301i·11-s + (−0.201 + 0.174i)12-s − 0.365i·13-s + (−0.0110 − 0.0297i)14-s + (−0.123 + 0.235i)15-s + (0.143 − 0.989i)16-s − 0.286i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.230i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $-0.973 + 0.230i$
Analytic conductor: \(5.99456\)
Root analytic conductor: \(2.44838\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :1),\ -0.973 + 0.230i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.122421 - 1.04848i\)
\(L(\frac12)\) \(\approx\) \(0.122421 - 1.04848i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.698 - 1.87i)T \)
5 \( 1 + (2.31 - 4.43i)T \)
11 \( 1 - 3.31iT \)
good3 \( 1 - 0.798T + 9T^{2} \)
7 \( 1 + 0.222T + 49T^{2} \)
13 \( 1 + 4.74iT - 169T^{2} \)
17 \( 1 + 4.87iT - 289T^{2} \)
19 \( 1 - 35.8iT - 361T^{2} \)
23 \( 1 + 2.86T + 529T^{2} \)
29 \( 1 + 12.9T + 841T^{2} \)
31 \( 1 - 34.8iT - 961T^{2} \)
37 \( 1 - 39.3iT - 1.36e3T^{2} \)
41 \( 1 - 51.7T + 1.68e3T^{2} \)
43 \( 1 - 44.8T + 1.84e3T^{2} \)
47 \( 1 - 28.0T + 2.20e3T^{2} \)
53 \( 1 - 0.246iT - 2.80e3T^{2} \)
59 \( 1 - 50.0iT - 3.48e3T^{2} \)
61 \( 1 + 77.8T + 3.72e3T^{2} \)
67 \( 1 - 107.T + 4.48e3T^{2} \)
71 \( 1 + 58.5iT - 5.04e3T^{2} \)
73 \( 1 - 84.8iT - 5.32e3T^{2} \)
79 \( 1 - 74.9iT - 6.24e3T^{2} \)
83 \( 1 - 118.T + 6.88e3T^{2} \)
89 \( 1 + 38.5T + 7.92e3T^{2} \)
97 \( 1 - 3.24iT - 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.61174304935243743961123215059, −11.82962845720794083680094437901, −10.62269872428733109725544079424, −9.468899615293090551670557578647, −8.233755388717407171686883107335, −7.63312324188658900247085432964, −6.45585845368145093339211283764, −5.51464146054469358930842658754, −3.96235642424518456297574414530, −2.92519775773711760217770662140, 0.49319211900431443884806113052, 2.41223248329212558846555768472, 3.80281381164776706647241601594, 4.89733078939604365153376500082, 6.00114005099320067634799861615, 7.80870106658886933914313985691, 8.976781378975942821525567704874, 9.324911264206499200765617605609, 11.00760463479028083650801241264, 11.43539168141309929731676402533

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