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Properties

Label 2-220-220.43-c2-0-43
Degree $2$
Conductor $220$
Sign $-0.348 + 0.937i$
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  + (−1.96 + 0.361i)2-s + (−2.82 + 2.82i)3-s + (3.73 − 1.42i)4-s + (3.27 − 3.77i)5-s + (4.54 − 6.58i)6-s + (−1.01 + 1.01i)7-s + (−6.84 + 4.14i)8-s − 6.99i·9-s + (−5.08 + 8.60i)10-s + (−9.38 + 5.74i)11-s + (−6.55 + 14.5i)12-s + (−6.58 + 6.58i)13-s + (1.62 − 2.36i)14-s + (1.40 + 19.9i)15-s + (11.9 − 10.6i)16-s + (−4.97 − 4.97i)17-s + ⋯
L(s)  = 1  + (−0.983 + 0.180i)2-s + (−0.942 + 0.942i)3-s + (0.934 − 0.355i)4-s + (0.655 − 0.754i)5-s + (0.756 − 1.09i)6-s + (−0.144 + 0.144i)7-s + (−0.855 + 0.518i)8-s − 0.777i·9-s + (−0.508 + 0.860i)10-s + (−0.852 + 0.522i)11-s + (−0.545 + 1.21i)12-s + (−0.506 + 0.506i)13-s + (0.116 − 0.168i)14-s + (0.0933 + 1.32i)15-s + (0.747 − 0.664i)16-s + (−0.292 − 0.292i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.348 + 0.937i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.348 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $-0.348 + 0.937i$
Analytic conductor: \(5.99456\)
Root analytic conductor: \(2.44838\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :1),\ -0.348 + 0.937i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.100183 - 0.144146i\)
\(L(\frac12)\) \(\approx\) \(0.100183 - 0.144146i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.96 - 0.361i)T \)
5 \( 1 + (-3.27 + 3.77i)T \)
11 \( 1 + (9.38 - 5.74i)T \)
good3 \( 1 + (2.82 - 2.82i)T - 9iT^{2} \)
7 \( 1 + (1.01 - 1.01i)T - 49iT^{2} \)
13 \( 1 + (6.58 - 6.58i)T - 169iT^{2} \)
17 \( 1 + (4.97 + 4.97i)T + 289iT^{2} \)
19 \( 1 + 28.5iT - 361T^{2} \)
23 \( 1 + (6.64 - 6.64i)T - 529iT^{2} \)
29 \( 1 + 9.10T + 841T^{2} \)
31 \( 1 + 34.1iT - 961T^{2} \)
37 \( 1 + (-0.442 + 0.442i)T - 1.36e3iT^{2} \)
41 \( 1 + 37.1iT - 1.68e3T^{2} \)
43 \( 1 + (-24.3 - 24.3i)T + 1.84e3iT^{2} \)
47 \( 1 + (38.0 + 38.0i)T + 2.20e3iT^{2} \)
53 \( 1 + (66.2 + 66.2i)T + 2.80e3iT^{2} \)
59 \( 1 + 83.9T + 3.48e3T^{2} \)
61 \( 1 + 8.08iT - 3.72e3T^{2} \)
67 \( 1 + (3.95 + 3.95i)T + 4.48e3iT^{2} \)
71 \( 1 - 42.6iT - 5.04e3T^{2} \)
73 \( 1 + (-93.4 + 93.4i)T - 5.32e3iT^{2} \)
79 \( 1 - 41.4iT - 6.24e3T^{2} \)
83 \( 1 + (65.0 + 65.0i)T + 6.88e3iT^{2} \)
89 \( 1 - 125. iT - 7.92e3T^{2} \)
97 \( 1 + (62.8 - 62.8i)T - 9.40e3iT^{2} \)
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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

−11.41114791344054044085236493825, −10.65142141259002556540733152249, −9.595572421997782518008797540968, −9.322376221988732555898933376848, −7.87159905942938886674393341910, −6.56424522230243653408423933412, −5.44298115091655408514319488272, −4.68748988020310492157987029677, −2.27944398703660691187197843713, −0.14091300601135852732477424548, 1.61571909499425724310316659564, 3.05789951024121841573603009789, 5.66364277159350772641617777300, 6.34506755477698986182416021090, 7.32591971220799704367548939254, 8.174215536403395792020557772123, 9.696430460635440019370243994327, 10.54453509351945072033217114689, 11.13213536256530780569954506891, 12.30449910018392140864552612669

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