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Properties

Label 2-220-44.43-c1-0-18
Degree $2$
Conductor $220$
Sign $0.511 + 0.859i$
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  + (1.32 − 0.5i)2-s − 0.613i·3-s + (1.50 − 1.32i)4-s − 5-s + (−0.306 − 0.811i)6-s − 0.613·7-s + (1.32 − 2.50i)8-s + 2.62·9-s + (−1.32 + 0.5i)10-s + (0.613 − 3.25i)11-s + (−0.811 − 0.920i)12-s + 2i·13-s + (−0.811 + 0.306i)14-s + 0.613i·15-s + (0.500 − 3.96i)16-s + 5.62i·17-s + ⋯
L(s)  = 1  + (0.935 − 0.353i)2-s − 0.354i·3-s + (0.750 − 0.661i)4-s − 0.447·5-s + (−0.125 − 0.331i)6-s − 0.231·7-s + (0.467 − 0.883i)8-s + 0.874·9-s + (−0.418 + 0.158i)10-s + (0.185 − 0.982i)11-s + (−0.234 − 0.265i)12-s + 0.554i·13-s + (−0.216 + 0.0819i)14-s + 0.158i·15-s + (0.125 − 0.992i)16-s + 1.36i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.511 + 0.859i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.511 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $0.511 + 0.859i$
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: \((2,\ 220,\ (\ :1/2),\ 0.511 + 0.859i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70129 - 0.967497i\)
\(L(\frac12)\) \(\approx\) \(1.70129 - 0.967497i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.32 + 0.5i)T \)
5 \( 1 + T \)
11 \( 1 + (-0.613 + 3.25i)T \)
good3 \( 1 + 0.613iT - 3T^{2} \)
7 \( 1 + 0.613T + 7T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - 5.62iT - 17T^{2} \)
19 \( 1 + 7.13T + 19T^{2} \)
23 \( 1 - 6.51iT - 23T^{2} \)
29 \( 1 - 7.62iT - 29T^{2} \)
31 \( 1 + 0.613iT - 31T^{2} \)
37 \( 1 - 5.62T + 37T^{2} \)
41 \( 1 + 8iT - 41T^{2} \)
43 \( 1 - 6.51T + 43T^{2} \)
47 \( 1 - 6.51iT - 47T^{2} \)
53 \( 1 + 9.62T + 53T^{2} \)
59 \( 1 - 4.06iT - 59T^{2} \)
61 \( 1 + 11.6iT - 61T^{2} \)
67 \( 1 + 7.74iT - 67T^{2} \)
71 \( 1 + 1.84iT - 71T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 - 2.45T + 79T^{2} \)
83 \( 1 + 7.74T + 83T^{2} \)
89 \( 1 - 1.62T + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−12.47733768249347090899845666569, −11.19244826077553528193975567696, −10.63611384162263134270081096427, −9.302583964145597720057735355320, −7.948416941633457857628758644878, −6.76160184448556674505396143968, −5.96954225189724917728549015422, −4.40838586209154441095090518196, −3.49951351285668897638779749280, −1.68299960504030342333200891621, 2.56109076747685554927755538732, 4.17876226032412850490795430184, 4.72864526896438551904392995787, 6.33060309320797108984141882001, 7.20503025843101358721975190896, 8.208519786980595707623025516098, 9.653101516169544190165587466547, 10.64064219934600006103879218635, 11.70772246995943408723413675741, 12.67415710629668566493853972517

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