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Properties

Label 2-220-11.2-c2-0-7
Degree $2$
Conductor $220$
Sign $0.171 + 0.985i$
Analytic cond. $5.99456$
Root an. cond. $2.44838$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.31 − 2.40i)3-s + (−0.690 − 2.12i)5-s + (3.59 − 4.95i)7-s + (2.39 − 7.37i)9-s + (−9.98 − 4.62i)11-s + (5.95 + 1.93i)13-s + (−7.40 − 5.37i)15-s + (12.0 − 3.91i)17-s + (1.94 + 2.67i)19-s − 25.0i·21-s − 15.1·23-s + (−4.04 + 2.93i)25-s + (1.57 + 4.86i)27-s + (3.04 − 4.18i)29-s + (1.12 − 3.46i)31-s + ⋯
L(s)  = 1  + (1.10 − 0.801i)3-s + (−0.138 − 0.425i)5-s + (0.514 − 0.707i)7-s + (0.266 − 0.819i)9-s + (−0.907 − 0.420i)11-s + (0.458 + 0.148i)13-s + (−0.493 − 0.358i)15-s + (0.708 − 0.230i)17-s + (0.102 + 0.140i)19-s − 1.19i·21-s − 0.658·23-s + (−0.161 + 0.117i)25-s + (0.0584 + 0.180i)27-s + (0.104 − 0.144i)29-s + (0.0363 − 0.111i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.171 + 0.985i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.171 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $0.171 + 0.985i$
Analytic conductor: \(5.99456\)
Root analytic conductor: \(2.44838\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :1),\ 0.171 + 0.985i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.64533 - 1.38354i\)
\(L(\frac12)\) \(\approx\) \(1.64533 - 1.38354i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.690 + 2.12i)T \)
11 \( 1 + (9.98 + 4.62i)T \)
good3 \( 1 + (-3.31 + 2.40i)T + (2.78 - 8.55i)T^{2} \)
7 \( 1 + (-3.59 + 4.95i)T + (-15.1 - 46.6i)T^{2} \)
13 \( 1 + (-5.95 - 1.93i)T + (136. + 99.3i)T^{2} \)
17 \( 1 + (-12.0 + 3.91i)T + (233. - 169. i)T^{2} \)
19 \( 1 + (-1.94 - 2.67i)T + (-111. + 343. i)T^{2} \)
23 \( 1 + 15.1T + 529T^{2} \)
29 \( 1 + (-3.04 + 4.18i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (-1.12 + 3.46i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (-58.1 - 42.2i)T + (423. + 1.30e3i)T^{2} \)
41 \( 1 + (9.53 + 13.1i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 + 42.6iT - 1.84e3T^{2} \)
47 \( 1 + (-26.4 + 19.2i)T + (682. - 2.10e3i)T^{2} \)
53 \( 1 + (13.7 - 42.2i)T + (-2.27e3 - 1.65e3i)T^{2} \)
59 \( 1 + (-41.8 - 30.3i)T + (1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (24.6 - 7.99i)T + (3.01e3 - 2.18e3i)T^{2} \)
67 \( 1 - 2.46T + 4.48e3T^{2} \)
71 \( 1 + (-30.8 - 95.0i)T + (-4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (85.0 - 117. i)T + (-1.64e3 - 5.06e3i)T^{2} \)
79 \( 1 + (-112. - 36.4i)T + (5.04e3 + 3.66e3i)T^{2} \)
83 \( 1 + (73.5 - 23.8i)T + (5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 + 139.T + 7.92e3T^{2} \)
97 \( 1 + (21.7 - 67.0i)T + (-7.61e3 - 5.53e3i)T^{2} \)
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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

−12.01772481218378968948802480337, −10.89071472509620553651903502864, −9.780926998634614157423662430408, −8.482348834559651278260233301950, −7.988883571177958406041247274660, −7.14180335779856342504025457249, −5.59009416082309897093933904005, −4.09503403416792068761678788522, −2.71089974741665984231585957991, −1.18200970479244797942842150445, 2.31668758197121040118678882664, 3.40301133812078036699417494964, 4.67428290399341980847504235858, 5.95788922597697799552314204981, 7.67483941261083086989547412559, 8.302526765890833847400836067464, 9.359114058291457889929959694933, 10.18022229174306059933380270733, 11.13396928965171396780971343827, 12.28302267247584980002809038502

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