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Properties

Label 2-22e2-11.5-c1-0-1
Degree $2$
Conductor $484$
Sign $0.957 + 0.288i$
Analytic cond. $3.86475$
Root an. cond. $1.96589$
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  + (−2.72 − 1.98i)3-s + (−0.424 + 1.30i)5-s + (2.58 + 7.96i)9-s + (3.74 − 2.72i)15-s + 7.37·23-s + (2.52 + 1.83i)25-s + (5.59 − 17.2i)27-s + (−1.89 − 5.81i)31-s + (9.80 − 7.12i)37-s − 11.4·45-s + (9.70 + 7.05i)47-s + (−2.16 + 6.65i)49-s + (1.85 + 5.70i)53-s + (−3.74 + 2.72i)59-s + 2.11·67-s + ⋯
L(s)  = 1  + (−1.57 − 1.14i)3-s + (−0.189 + 0.583i)5-s + (0.862 + 2.65i)9-s + (0.966 − 0.702i)15-s + 1.53·23-s + (0.504 + 0.366i)25-s + (1.07 − 3.31i)27-s + (−0.339 − 1.04i)31-s + (1.61 − 1.17i)37-s − 1.71·45-s + (1.41 + 1.02i)47-s + (−0.309 + 0.951i)49-s + (0.254 + 0.783i)53-s + (−0.487 + 0.354i)59-s + 0.258·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 + 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(484\)    =    \(2^{2} \cdot 11^{2}\)
Sign: $0.957 + 0.288i$
Analytic conductor: \(3.86475\)
Root analytic conductor: \(1.96589\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{484} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 484,\ (\ :1/2),\ 0.957 + 0.288i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.770766 - 0.113406i\)
\(L(\frac12)\) \(\approx\) \(0.770766 - 0.113406i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
good3 \( 1 + (2.72 + 1.98i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (0.424 - 1.30i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 7.37T + 23T^{2} \)
29 \( 1 + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.89 + 5.81i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-9.80 + 7.12i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (-9.70 - 7.05i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-1.85 - 5.70i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (3.74 - 2.72i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 2.11T + 67T^{2} \)
71 \( 1 + (3.97 - 12.2i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 18.8T + 89T^{2} \)
97 \( 1 + (-0.0361 - 0.111i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.06587429925956427347225805042, −10.56774131527294145894093690009, −9.201011051267406183907763633551, −7.68300520109498940199231326199, −7.24464645613460429257225795935, −6.29458155391375491829183168705, −5.56661375005930701179822302251, −4.45195318940108000079053385903, −2.56519760998494237081996228583, −0.982326731779330030005975931107, 0.821565767192406501941926532783, 3.43965025501253959726625461968, 4.64342302185092712967804940906, 5.10500495554074342971236838362, 6.17497231280830151088830888605, 7.06194554164863571309276837262, 8.636494178954520980388712639783, 9.416179827252748620126900252187, 10.31719491151938519955285018400, 10.98203638488499886049757248508

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