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Properties

Label 2-220-44.7-c1-0-6
Degree $2$
Conductor $220$
Sign $0.233 + 0.972i$
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  + (−1.26 − 0.627i)2-s + (−1.88 − 0.612i)3-s + (1.21 + 1.59i)4-s + (0.809 − 0.587i)5-s + (2.00 + 1.96i)6-s + (0.668 + 2.05i)7-s + (−0.539 − 2.77i)8-s + (0.755 + 0.548i)9-s + (−1.39 + 0.237i)10-s + (3.31 + 0.105i)11-s + (−1.31 − 3.74i)12-s + (1.85 − 2.55i)13-s + (0.443 − 3.02i)14-s + (−1.88 + 0.612i)15-s + (−1.05 + 3.85i)16-s + (−2.62 − 3.60i)17-s + ⋯
L(s)  = 1  + (−0.896 − 0.443i)2-s + (−1.08 − 0.353i)3-s + (0.606 + 0.795i)4-s + (0.361 − 0.262i)5-s + (0.819 + 0.800i)6-s + (0.252 + 0.777i)7-s + (−0.190 − 0.981i)8-s + (0.251 + 0.182i)9-s + (−0.440 + 0.0750i)10-s + (0.999 + 0.0316i)11-s + (−0.379 − 1.08i)12-s + (0.515 − 0.709i)13-s + (0.118 − 0.809i)14-s + (−0.487 + 0.158i)15-s + (−0.264 + 0.964i)16-s + (−0.635 − 0.874i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.233 + 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.233 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $0.233 + 0.972i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :1/2),\ 0.233 + 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.488553 - 0.384918i\)
\(L(\frac12)\) \(\approx\) \(0.488553 - 0.384918i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.26 + 0.627i)T \)
5 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (-3.31 - 0.105i)T \)
good3 \( 1 + (1.88 + 0.612i)T + (2.42 + 1.76i)T^{2} \)
7 \( 1 + (-0.668 - 2.05i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-1.85 + 2.55i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (2.62 + 3.60i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.21 + 3.74i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 8.67iT - 23T^{2} \)
29 \( 1 + (-6.12 + 1.99i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.673 - 0.927i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (-3.17 - 9.78i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.29 + 0.421i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + 4.00T + 43T^{2} \)
47 \( 1 + (-3.61 - 1.17i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-6.21 - 4.51i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.112 + 0.0366i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (7.07 + 9.74i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 - 6.69iT - 67T^{2} \)
71 \( 1 + (0.556 + 0.766i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.45 + 0.471i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (3.49 + 2.53i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-0.790 + 0.574i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 13.2T + 89T^{2} \)
97 \( 1 + (-11.7 - 8.52i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.89010530603572360982992662835, −11.25176303146210697988742531117, −10.26914629483125830528139178925, −9.066670993631965027493072274439, −8.436078263932833474915328917093, −6.83115657455616305474415272960, −6.17761854652206348656129562925, −4.76960222343025091658111338697, −2.69756038019811534428764253522, −0.907462102749955900493532762874, 1.47679682169807218339312834579, 4.08934291552501013696870829348, 5.59633662589896588083797546851, 6.36305735165699400323632235048, 7.31666539140329952540481791194, 8.644926225352095163268942746399, 9.696558419705093250784650684264, 10.58996349938243587182017056071, 11.22133367057719886449893381888, 11.98134171073134742591159952258

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