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Properties

Label 2-220-5.2-c2-0-9
Degree $2$
Conductor $220$
Sign $-0.917 + 0.398i$
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  + (−0.176 + 0.176i)3-s + (0.718 − 4.94i)5-s + (−8.33 − 8.33i)7-s + 8.93i·9-s − 3.31·11-s + (−17.8 + 17.8i)13-s + (0.746 + 0.999i)15-s + (−14.0 − 14.0i)17-s − 14.2i·19-s + 2.94·21-s + (17.1 − 17.1i)23-s + (−23.9 − 7.10i)25-s + (−3.16 − 3.16i)27-s − 9.52i·29-s + 17.4·31-s + ⋯
L(s)  = 1  + (−0.0588 + 0.0588i)3-s + (0.143 − 0.989i)5-s + (−1.19 − 1.19i)7-s + 0.993i·9-s − 0.301·11-s + (−1.37 + 1.37i)13-s + (0.0497 + 0.0666i)15-s + (−0.827 − 0.827i)17-s − 0.749i·19-s + 0.140·21-s + (0.745 − 0.745i)23-s + (−0.958 − 0.284i)25-s + (−0.117 − 0.117i)27-s − 0.328i·29-s + 0.561·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.917 + 0.398i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.917 + 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $-0.917 + 0.398i$
Analytic conductor: \(5.99456\)
Root analytic conductor: \(2.44838\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :1),\ -0.917 + 0.398i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.111721 - 0.538109i\)
\(L(\frac12)\) \(\approx\) \(0.111721 - 0.538109i\)
\(L(2)\) 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 \( 1 + (-0.718 + 4.94i)T \)
11 \( 1 + 3.31T \)
good3 \( 1 + (0.176 - 0.176i)T - 9iT^{2} \)
7 \( 1 + (8.33 + 8.33i)T + 49iT^{2} \)
13 \( 1 + (17.8 - 17.8i)T - 169iT^{2} \)
17 \( 1 + (14.0 + 14.0i)T + 289iT^{2} \)
19 \( 1 + 14.2iT - 361T^{2} \)
23 \( 1 + (-17.1 + 17.1i)T - 529iT^{2} \)
29 \( 1 + 9.52iT - 841T^{2} \)
31 \( 1 - 17.4T + 961T^{2} \)
37 \( 1 + (15.8 + 15.8i)T + 1.36e3iT^{2} \)
41 \( 1 + 40.5T + 1.68e3T^{2} \)
43 \( 1 + (-26.3 + 26.3i)T - 1.84e3iT^{2} \)
47 \( 1 + (-19.5 - 19.5i)T + 2.20e3iT^{2} \)
53 \( 1 + (-25.7 + 25.7i)T - 2.80e3iT^{2} \)
59 \( 1 - 24.2iT - 3.48e3T^{2} \)
61 \( 1 - 77.6T + 3.72e3T^{2} \)
67 \( 1 + (68.1 + 68.1i)T + 4.48e3iT^{2} \)
71 \( 1 - 28.0T + 5.04e3T^{2} \)
73 \( 1 + (1.22 - 1.22i)T - 5.32e3iT^{2} \)
79 \( 1 + 127. iT - 6.24e3T^{2} \)
83 \( 1 + (64.3 - 64.3i)T - 6.88e3iT^{2} \)
89 \( 1 - 125. iT - 7.92e3T^{2} \)
97 \( 1 + (25.6 + 25.6i)T + 9.40e3iT^{2} \)
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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

−11.69549631069812987668484696235, −10.53343723587865378408380036985, −9.667978947830502891821468892125, −8.875909963959189587043908866140, −7.40822400235407898023688984915, −6.72496719678979424413414156595, −5.00052219434141497326264898737, −4.32181250873677052903059426616, −2.41352739182780778672229813905, −0.28106233884226743977471888162, 2.59077708475202347540553384202, 3.42420094505077448539743587870, 5.49555824800514863130660829985, 6.30722495969737343789055340827, 7.25902842967036021110491370762, 8.644025451142753630943138114588, 9.761207458260218949542535672457, 10.29438564479458356619804454834, 11.66493028900231651552554760636, 12.52574111516389284231544908018

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