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Properties

Label 2-220-44.7-c1-0-10
Degree $2$
Conductor $220$
Sign $0.446 - 0.894i$
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  + (1.25 + 0.659i)2-s + (0.260 + 0.0846i)3-s + (1.13 + 1.64i)4-s + (−0.809 + 0.587i)5-s + (0.269 + 0.277i)6-s + (0.560 + 1.72i)7-s + (0.326 + 2.80i)8-s + (−2.36 − 1.71i)9-s + (−1.39 + 0.201i)10-s + (2.57 + 2.08i)11-s + (0.154 + 0.525i)12-s + (2.98 − 4.10i)13-s + (−0.436 + 2.52i)14-s + (−0.260 + 0.0846i)15-s + (−1.44 + 3.73i)16-s + (−2.63 − 3.62i)17-s + ⋯
L(s)  = 1  + (0.884 + 0.466i)2-s + (0.150 + 0.0488i)3-s + (0.565 + 0.824i)4-s + (−0.361 + 0.262i)5-s + (0.110 + 0.113i)6-s + (0.211 + 0.651i)7-s + (0.115 + 0.993i)8-s + (−0.788 − 0.573i)9-s + (−0.442 + 0.0638i)10-s + (0.777 + 0.628i)11-s + (0.0446 + 0.151i)12-s + (0.826 − 1.13i)13-s + (−0.116 + 0.675i)14-s + (−0.0672 + 0.0218i)15-s + (−0.361 + 0.932i)16-s + (−0.638 − 0.878i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.446 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.446 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $0.446 - 0.894i$
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: \((2,\ 220,\ (\ :1/2),\ 0.446 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65939 + 1.02688i\)
\(L(\frac12)\) \(\approx\) \(1.65939 + 1.02688i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.25 - 0.659i)T \)
5 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 + (-2.57 - 2.08i)T \)
good3 \( 1 + (-0.260 - 0.0846i)T + (2.42 + 1.76i)T^{2} \)
7 \( 1 + (-0.560 - 1.72i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-2.98 + 4.10i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (2.63 + 3.62i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.882 + 2.71i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 0.353iT - 23T^{2} \)
29 \( 1 + (3.78 - 1.23i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.598 + 0.823i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.34 + 4.14i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-3.62 - 1.17i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + 11.9T + 43T^{2} \)
47 \( 1 + (-10.2 - 3.33i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (4.99 + 3.62i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (9.49 - 3.08i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-7.97 - 10.9i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + 12.6iT - 67T^{2} \)
71 \( 1 + (-1.91 - 2.63i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (1.42 - 0.463i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-10.3 - 7.53i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (8.34 - 6.06i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 1.53T + 89T^{2} \)
97 \( 1 + (6.18 + 4.49i)T + (29.9 + 92.2i)T^{2} \)
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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

−12.44577980052329479530836231802, −11.66985993888688313984734181253, −10.96762955792088112208003628844, −9.232018292442012515169845979086, −8.404207001207297295923907706367, −7.24291256864793866757943970797, −6.21086329562625137044824442421, −5.18522500088263977453871355809, −3.79913737038140130758222057957, −2.67660069987443737981825012533, 1.66012849052636055620186476570, 3.53070058255571875370886638314, 4.36692741522658801330528594410, 5.78203634051181266189329167156, 6.77314819783482800974989278855, 8.179447643466625187645474077777, 9.187454769551471860805920017183, 10.63320319370674703827999592745, 11.28832280858181430976383323980, 11.98368925734759327139776405396

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