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Properties

Label 2-220-220.219-c1-0-19
Degree $2$
Conductor $220$
Sign $0.205 + 0.978i$
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  + (−1.33 − 0.478i)2-s + 1.30·3-s + (1.54 + 1.27i)4-s + (−0.393 − 2.20i)5-s + (−1.73 − 0.624i)6-s − 1.57i·7-s + (−1.44 − 2.43i)8-s − 1.29·9-s + (−0.529 + 3.11i)10-s + (2.49 − 2.18i)11-s + (2.01 + 1.66i)12-s + 1.50·13-s + (−0.755 + 2.09i)14-s + (−0.513 − 2.87i)15-s + (0.755 + 3.92i)16-s − 4.38·17-s + ⋯
L(s)  = 1  + (−0.941 − 0.338i)2-s + 0.753·3-s + (0.770 + 0.636i)4-s + (−0.175 − 0.984i)5-s + (−0.708 − 0.254i)6-s − 0.596i·7-s + (−0.509 − 0.860i)8-s − 0.432·9-s + (−0.167 + 0.985i)10-s + (0.751 − 0.659i)11-s + (0.580 + 0.479i)12-s + 0.416·13-s + (−0.201 + 0.561i)14-s + (−0.132 − 0.741i)15-s + (0.188 + 0.982i)16-s − 1.06·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.205 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.205 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $0.205 + 0.978i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :1/2),\ 0.205 + 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.751698 - 0.610359i\)
\(L(\frac12)\) \(\approx\) \(0.751698 - 0.610359i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.33 + 0.478i)T \)
5 \( 1 + (0.393 + 2.20i)T \)
11 \( 1 + (-2.49 + 2.18i)T \)
good3 \( 1 - 1.30T + 3T^{2} \)
7 \( 1 + 1.57iT - 7T^{2} \)
13 \( 1 - 1.50T + 13T^{2} \)
17 \( 1 + 4.38T + 17T^{2} \)
19 \( 1 - 5.72T + 19T^{2} \)
23 \( 1 - 5.32T + 23T^{2} \)
29 \( 1 + 6.34iT - 29T^{2} \)
31 \( 1 + 4.48iT - 31T^{2} \)
37 \( 1 - 6.78iT - 37T^{2} \)
41 \( 1 - 8.40iT - 41T^{2} \)
43 \( 1 - 11.5iT - 43T^{2} \)
47 \( 1 - 4.30T + 47T^{2} \)
53 \( 1 + 2.38iT - 53T^{2} \)
59 \( 1 - 7.74iT - 59T^{2} \)
61 \( 1 - 4.55iT - 61T^{2} \)
67 \( 1 + 5.32T + 67T^{2} \)
71 \( 1 - 6.85iT - 71T^{2} \)
73 \( 1 + 2.63T + 73T^{2} \)
79 \( 1 + 4.98T + 79T^{2} \)
83 \( 1 + 1.78iT - 83T^{2} \)
89 \( 1 + 2.72T + 89T^{2} \)
97 \( 1 + 4.40iT - 97T^{2} \)
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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

−11.63834132137604500236834667155, −11.32424079482241075067570187389, −9.785407007206793221669777938478, −9.037095386002341257720084642386, −8.382782450515358549892857440964, −7.48153936028077550895555166582, −6.09153184365118856609519761181, −4.21367371859033623376301866996, −2.97490051431489866522112684772, −1.10464868421829108742534374899, 2.16166842567365339613738215129, 3.36602818591643755767482445815, 5.51034168669342553732008438524, 6.80431005865181383801167615863, 7.45486686862180651099447096056, 8.890647366156406703842207693136, 9.102366899684564633385417326866, 10.48998958704110463185426712149, 11.29073370875188108663350789451, 12.19778735797324866299201688756

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