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Properties

Label 2-22e2-11.5-c1-0-7
Degree $2$
Conductor $484$
Sign $-0.598 + 0.801i$
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  + (−1.61 − 1.17i)3-s + (0.927 − 2.85i)5-s + (2.80 − 2.03i)7-s + (0.309 + 0.951i)9-s + (1.60 + 4.94i)13-s + (−4.85 + 3.52i)15-s + (1.60 − 4.94i)17-s + (−2.80 − 2.03i)19-s − 6.92·21-s − 6·23-s + (−3.23 − 2.35i)25-s + (−1.23 + 3.80i)27-s + (4.20 − 3.05i)29-s + (−0.618 − 1.90i)31-s + (−3.21 − 9.88i)35-s + ⋯
L(s)  = 1  + (−0.934 − 0.678i)3-s + (0.414 − 1.27i)5-s + (1.05 − 0.769i)7-s + (0.103 + 0.317i)9-s + (0.445 + 1.37i)13-s + (−1.25 + 0.910i)15-s + (0.389 − 1.19i)17-s + (−0.642 − 0.467i)19-s − 1.51·21-s − 1.25·23-s + (−0.647 − 0.470i)25-s + (−0.237 + 0.732i)27-s + (0.780 − 0.567i)29-s + (−0.111 − 0.341i)31-s + (−0.542 − 1.67i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.598 + 0.801i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.598 + 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(484\)    =    \(2^{2} \cdot 11^{2}\)
Sign: $-0.598 + 0.801i$
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: \((2,\ 484,\ (\ :1/2),\ -0.598 + 0.801i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.509798 - 1.01689i\)
\(L(\frac12)\) \(\approx\) \(0.509798 - 1.01689i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.61 + 1.17i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (-0.927 + 2.85i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-2.80 + 2.03i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-1.60 - 4.94i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.60 + 4.94i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (2.80 + 2.03i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 + (-4.20 + 3.05i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.618 + 1.90i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (0.809 - 0.587i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-4.20 - 3.05i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (4.85 + 3.52i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (0.927 + 2.85i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (4.28 - 13.1i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 2T + 67T^{2} \)
71 \( 1 + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-5.60 + 4.07i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.07 - 3.29i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (3.21 - 9.88i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 15T + 89T^{2} \)
97 \( 1 + (-1.54 - 4.75i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.95470132452959813344955082962, −9.725303744007604872938571280779, −8.846371507336898512007308789703, −7.900655758900141240040269421004, −6.89355361682402391650804334200, −5.97214711796449226311381075236, −4.92597184195358740126527584761, −4.26247621353726869310390610888, −1.83256190467801924892945253067, −0.817366450672638869823767492225, 2.04516859980176560327965333176, 3.44874319110192882699528806931, 4.80620361456691607595968796910, 5.84955439721423124083975158313, 6.19366212223932540897325454231, 7.79351098054135579557806811153, 8.470252684292407245976197395756, 9.988964955633032890889426837123, 10.61736141390583614379396534016, 10.92737299881424813387332705731

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