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Properties

Label 2-220-44.43-c1-0-20
Degree $2$
Conductor $220$
Sign $0.260 + 0.965i$
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  + (1.39 − 0.249i)2-s − 2.27i·3-s + (1.87 − 0.695i)4-s + 5-s + (−0.567 − 3.16i)6-s − 3.82·7-s + (2.43 − 1.43i)8-s − 2.15·9-s + (1.39 − 0.249i)10-s + (−0.302 + 3.30i)11-s + (−1.57 − 4.25i)12-s + 4.63i·13-s + (−5.32 + 0.956i)14-s − 2.27i·15-s + (3.03 − 2.60i)16-s − 0.686i·17-s + ⋯
L(s)  = 1  + (0.984 − 0.176i)2-s − 1.31i·3-s + (0.937 − 0.347i)4-s + 0.447·5-s + (−0.231 − 1.29i)6-s − 1.44·7-s + (0.861 − 0.507i)8-s − 0.719·9-s + (0.440 − 0.0789i)10-s + (−0.0911 + 0.995i)11-s + (−0.455 − 1.22i)12-s + 1.28i·13-s + (−1.42 + 0.255i)14-s − 0.586i·15-s + (0.758 − 0.651i)16-s − 0.166i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.260 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.260 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $0.260 + 0.965i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :1/2),\ 0.260 + 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.63384 - 1.25120i\)
\(L(\frac12)\) \(\approx\) \(1.63384 - 1.25120i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 + 0.249i)T \)
5 \( 1 - T \)
11 \( 1 + (0.302 - 3.30i)T \)
good3 \( 1 + 2.27iT - 3T^{2} \)
7 \( 1 + 3.82T + 7T^{2} \)
13 \( 1 - 4.63iT - 13T^{2} \)
17 \( 1 + 0.686iT - 17T^{2} \)
19 \( 1 - 3.82T + 19T^{2} \)
23 \( 1 + 4.18iT - 23T^{2} \)
29 \( 1 + 3.95iT - 29T^{2} \)
31 \( 1 - 4.69iT - 31T^{2} \)
37 \( 1 + 6.34T + 37T^{2} \)
41 \( 1 - 6.74iT - 41T^{2} \)
43 \( 1 + 2.87T + 43T^{2} \)
47 \( 1 - 12.5iT - 47T^{2} \)
53 \( 1 + 8.65T + 53T^{2} \)
59 \( 1 - 1.76iT - 59T^{2} \)
61 \( 1 + 5.32iT - 61T^{2} \)
67 \( 1 + 4.48iT - 67T^{2} \)
71 \( 1 - 0.868iT - 71T^{2} \)
73 \( 1 + 16.1iT - 73T^{2} \)
79 \( 1 + 16.9T + 79T^{2} \)
83 \( 1 + 4.78T + 83T^{2} \)
89 \( 1 - 5.47T + 89T^{2} \)
97 \( 1 - 17.6T + 97T^{2} \)
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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

−12.43648395423572870459701004852, −11.64103331420624860659926810317, −10.18866428166458405848414956999, −9.357070496291576976293074444836, −7.53495975805134847675627405358, −6.67996324522392562247381617590, −6.24006426723530290361926395734, −4.63994657498015817772683186238, −2.99201337623790660238674931330, −1.74571274180383667961085625767, 3.12752877759680587859872222956, 3.63202919247051812961959745631, 5.29341600032119601072063398567, 5.84202998758659684397507809962, 7.16873502632218701495343664776, 8.699964952455338479617100883210, 9.910224919446315437882780054099, 10.42894991252824074832968314788, 11.49776970444422632388312337601, 12.77305226823515875891606318456

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