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Properties

Label 2-22e2-11.4-c1-0-4
Degree $2$
Conductor $484$
Sign $0.995 - 0.0913i$
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  + (−0.118 + 0.363i)3-s + (−0.5 + 0.363i)5-s + (−0.881 − 2.71i)7-s + (2.30 + 1.67i)9-s + (3.73 + 2.71i)13-s + (−0.0729 − 0.224i)15-s + (3.73 − 2.71i)17-s + (0.881 − 2.71i)19-s + 1.09·21-s + 6.47·23-s + (−1.42 + 4.39i)25-s + (−1.80 + 1.31i)27-s + (−1.97 − 6.06i)29-s + (4.73 + 3.44i)31-s + (1.42 + 1.03i)35-s + ⋯
L(s)  = 1  + (−0.0681 + 0.209i)3-s + (−0.223 + 0.162i)5-s + (−0.333 − 1.02i)7-s + (0.769 + 0.559i)9-s + (1.03 + 0.752i)13-s + (−0.0188 − 0.0579i)15-s + (0.906 − 0.658i)17-s + (0.202 − 0.622i)19-s + 0.237·21-s + 1.34·23-s + (−0.285 + 0.878i)25-s + (−0.348 + 0.252i)27-s + (−0.366 − 1.12i)29-s + (0.850 + 0.618i)31-s + (0.241 + 0.175i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(484\)    =    \(2^{2} \cdot 11^{2}\)
Sign: $0.995 - 0.0913i$
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: \((2,\ 484,\ (\ :1/2),\ 0.995 - 0.0913i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44558 + 0.0661955i\)
\(L(\frac12)\) \(\approx\) \(1.44558 + 0.0661955i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.118 - 0.363i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (0.5 - 0.363i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (0.881 + 2.71i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-3.73 - 2.71i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-3.73 + 2.71i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.881 + 2.71i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 6.47T + 23T^{2} \)
29 \( 1 + (1.97 + 6.06i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-4.73 - 3.44i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.5 - 4.61i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.97 - 6.06i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (-1.11 + 3.44i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (5.73 + 4.16i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (3.11 + 9.59i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-3.73 + 2.71i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 4.94T + 67T^{2} \)
71 \( 1 + (-6.97 + 5.06i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-3.73 - 11.4i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (11.2 + 8.14i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (12.9 - 9.42i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 8.47T + 89T^{2} \)
97 \( 1 + (4.5 + 3.26i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.07659586413963802115951150906, −10.05479482785589625140908927988, −9.454117835753533913635340927329, −8.181289468019740592523269625308, −7.23667885501764901745927886175, −6.61594517305613766809488608179, −5.14247071291650841439004644711, −4.18480666420232805795138244021, −3.18126329673834939276511521875, −1.24518582605777636966553691239, 1.25393392761175148878145354219, 2.99799713131858320269564474451, 4.05283133593475884171647846471, 5.54231661563834750621789974024, 6.17518770396525502728439652262, 7.34728901870440665801905508973, 8.338327568235790360178298219577, 9.109302610406789241895403802419, 10.07391323109563137019186566880, 10.94249800104696407319489224162

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