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Properties

Label 2-220-20.3-c1-0-18
Degree $2$
Conductor $220$
Sign $0.999 - 0.0120i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.41 − 0.00742i)2-s + (0.373 − 0.373i)3-s + (1.99 − 0.0210i)4-s + (1.17 + 1.90i)5-s + (0.525 − 0.530i)6-s + (−2.76 − 2.76i)7-s + (2.82 − 0.0445i)8-s + 2.72i·9-s + (1.68 + 2.67i)10-s + i·11-s + (0.739 − 0.754i)12-s + (−3.64 − 3.64i)13-s + (−3.92 − 3.88i)14-s + (1.14 + 0.269i)15-s + (3.99 − 0.0840i)16-s + (1.81 − 1.81i)17-s + ⋯
L(s)  = 1  + (0.999 − 0.00525i)2-s + (0.215 − 0.215i)3-s + (0.999 − 0.0105i)4-s + (0.527 + 0.849i)5-s + (0.214 − 0.216i)6-s + (−1.04 − 1.04i)7-s + (0.999 − 0.0157i)8-s + 0.907i·9-s + (0.531 + 0.847i)10-s + 0.301i·11-s + (0.213 − 0.217i)12-s + (−1.01 − 1.01i)13-s + (−1.04 − 1.03i)14-s + (0.296 + 0.0695i)15-s + (0.999 − 0.0210i)16-s + (0.440 − 0.440i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0120i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0120i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $0.999 - 0.0120i$
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: \((2,\ 220,\ (\ :1/2),\ 0.999 - 0.0120i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.22522 + 0.0134504i\)
\(L(\frac12)\) \(\approx\) \(2.22522 + 0.0134504i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 + 0.00742i)T \)
5 \( 1 + (-1.17 - 1.90i)T \)
11 \( 1 - iT \)
good3 \( 1 + (-0.373 + 0.373i)T - 3iT^{2} \)
7 \( 1 + (2.76 + 2.76i)T + 7iT^{2} \)
13 \( 1 + (3.64 + 3.64i)T + 13iT^{2} \)
17 \( 1 + (-1.81 + 1.81i)T - 17iT^{2} \)
19 \( 1 + 4.29T + 19T^{2} \)
23 \( 1 + (0.0710 - 0.0710i)T - 23iT^{2} \)
29 \( 1 - 2.00iT - 29T^{2} \)
31 \( 1 + 5.16iT - 31T^{2} \)
37 \( 1 + (-2.56 + 2.56i)T - 37iT^{2} \)
41 \( 1 + 5.55T + 41T^{2} \)
43 \( 1 + (6.95 - 6.95i)T - 43iT^{2} \)
47 \( 1 + (-6.82 - 6.82i)T + 47iT^{2} \)
53 \( 1 + (6.31 + 6.31i)T + 53iT^{2} \)
59 \( 1 + 3.52T + 59T^{2} \)
61 \( 1 - 10.1T + 61T^{2} \)
67 \( 1 + (-2.03 - 2.03i)T + 67iT^{2} \)
71 \( 1 - 2.60iT - 71T^{2} \)
73 \( 1 + (-3.23 - 3.23i)T + 73iT^{2} \)
79 \( 1 - 5.62T + 79T^{2} \)
83 \( 1 + (-2.63 + 2.63i)T - 83iT^{2} \)
89 \( 1 + 8.67iT - 89T^{2} \)
97 \( 1 + (-12.7 + 12.7i)T - 97iT^{2} \)
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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

−12.83185292619047937817718601670, −11.30826384910503044313791277827, −10.32788101153553398960753975789, −9.926491913417958112154603189552, −7.75101834697557697711531475813, −7.12415315126340381503844193903, −6.14827578222632260452669118881, −4.87046505866635540448624480254, −3.39031215097116196672169839289, −2.36641387220949498164034886444, 2.22771130520291193550428187627, 3.60042951001474505129146073287, 4.89901984968193412113485051548, 6.02262806281102327845303017498, 6.72141644350886702871901483088, 8.521210076503837615459004480424, 9.383678070803018891737363864669, 10.23994324965805698465459425390, 11.98266443217945602061223399785, 12.24170551714720859520817218456

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