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Properties

Label 2-22e2-11.8-c0-0-0
Degree $2$
Conductor $484$
Sign $0.794 + 0.606i$
Analytic cond. $0.241547$
Root an. cond. $0.491474$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.309 − 0.951i)3-s + (0.809 + 0.587i)5-s + (0.309 − 0.951i)15-s − 23-s + (−0.809 − 0.587i)27-s + (0.809 − 0.587i)31-s + (−0.309 + 0.951i)37-s + (0.618 + 1.90i)47-s + (−0.809 − 0.587i)49-s + (−1.61 + 1.17i)53-s + (−0.309 + 0.951i)59-s − 67-s + (0.309 + 0.951i)69-s + (0.809 + 0.587i)71-s + (−0.309 + 0.951i)81-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)3-s + (0.809 + 0.587i)5-s + (0.309 − 0.951i)15-s − 23-s + (−0.809 − 0.587i)27-s + (0.809 − 0.587i)31-s + (−0.309 + 0.951i)37-s + (0.618 + 1.90i)47-s + (−0.809 − 0.587i)49-s + (−1.61 + 1.17i)53-s + (−0.309 + 0.951i)59-s − 67-s + (0.309 + 0.951i)69-s + (0.809 + 0.587i)71-s + (−0.309 + 0.951i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(484\)    =    \(2^{2} \cdot 11^{2}\)
Sign: $0.794 + 0.606i$
Analytic conductor: \(0.241547\)
Root analytic conductor: \(0.491474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{484} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 484,\ (\ :0),\ 0.794 + 0.606i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9027912289\)
\(L(\frac12)\) \(\approx\) \(0.9027912289\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
good3 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
5 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
7 \( 1 + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (-0.309 + 0.951i)T^{2} \)
17 \( 1 + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + (0.809 - 0.587i)T^{2} \)
23 \( 1 + T + T^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.618 - 1.90i)T + (-0.809 + 0.587i)T^{2} \)
53 \( 1 + (1.61 - 1.17i)T + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.309 - 0.951i)T^{2} \)
67 \( 1 + T + T^{2} \)
71 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.309 - 0.951i)T^{2} \)
89 \( 1 + T + T^{2} \)
97 \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \)
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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

−11.17781934606120849553658893118, −10.16380323900371934251853598420, −9.523781187740648617609432445510, −8.201607329108578541326460875942, −7.34080915966402764168888150927, −6.35157331998815333512072655833, −5.93344240792411486147285153270, −4.40591490889285631995979424099, −2.80889034985975438213310779383, −1.59824617081529532245191527520, 1.89200849547520895737751663171, 3.62378668320487869449544313206, 4.75004823144366322880015998050, 5.44230536774242038934820007856, 6.45492897556092093585860188927, 7.78060751724439748216174708987, 8.872237741364514067790081734970, 9.665887322056624779139265465229, 10.24255454570569208788446252656, 11.09682677356640825439867445451

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