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Properties

Label 2-22e2-11.4-c1-0-5
Degree $2$
Conductor $484$
Sign $0.859 + 0.511i$
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  + (0.309 − 0.951i)3-s + (2.42 − 1.76i)5-s + (0.618 + 1.90i)7-s + (1.61 + 1.17i)9-s + (3.23 + 2.35i)13-s + (−0.927 − 2.85i)15-s + (−4.85 + 3.52i)17-s + (2.47 − 7.60i)19-s + 1.99·21-s − 3·23-s + (1.23 − 3.80i)25-s + (4.04 − 2.93i)27-s + (−4.04 − 2.93i)31-s + (4.85 + 3.52i)35-s + (−0.309 − 0.951i)37-s + ⋯
L(s)  = 1  + (0.178 − 0.549i)3-s + (1.08 − 0.788i)5-s + (0.233 + 0.718i)7-s + (0.539 + 0.391i)9-s + (0.897 + 0.652i)13-s + (−0.239 − 0.736i)15-s + (−1.17 + 0.855i)17-s + (0.567 − 1.74i)19-s + 0.436·21-s − 0.625·23-s + (0.247 − 0.760i)25-s + (0.778 − 0.565i)27-s + (−0.726 − 0.527i)31-s + (0.820 + 0.596i)35-s + (−0.0508 − 0.156i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(484\)    =    \(2^{2} \cdot 11^{2}\)
Sign: $0.859 + 0.511i$
Analytic conductor: \(3.86475\)
Root analytic conductor: \(1.96589\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{484} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 484,\ (\ :1/2),\ 0.859 + 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83726 - 0.505312i\)
\(L(\frac12)\) \(\approx\) \(1.83726 - 0.505312i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.309 + 0.951i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (-2.42 + 1.76i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-0.618 - 1.90i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-3.23 - 2.35i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (4.85 - 3.52i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-2.47 + 7.60i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 3T + 23T^{2} \)
29 \( 1 + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (4.04 + 2.93i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.309 + 0.951i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 10T + 43T^{2} \)
47 \( 1 + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-4.85 - 3.52i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.927 - 2.85i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-3.23 + 2.35i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + T + 67T^{2} \)
71 \( 1 + (12.1 - 8.81i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.23 + 3.80i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (1.61 + 1.17i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (4.85 - 3.52i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 9T + 89T^{2} \)
97 \( 1 + (-5.66 - 4.11i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.96913195243310255570109230756, −9.853193362698060389153447741812, −8.908246444180622716778263739857, −8.531874176159357273642441037417, −7.14550738462498947983188453177, −6.24642046996043666440156542584, −5.29800929212182368046779807246, −4.28558475966085868907472181800, −2.34280715172321585954235310915, −1.53830943892934459824734299309, 1.62456130286410344766333643030, 3.18947072054696784739710548672, 4.13393857707114012160221591251, 5.47023187261368888480374159416, 6.43577213230103727673694738799, 7.27493141054378902439361307788, 8.470322155529119781174001311838, 9.553268370836884402053352607643, 10.22930803244131686826743987196, 10.67226782146919687643869708313

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