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Properties

Label 2-1452-11.4-c1-0-11
Degree $2$
Conductor $1452$
Sign $0.859 + 0.511i$
Analytic cond. $11.5942$
Root an. cond. $3.40503$
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 + (3.23 − 2.35i)5-s + (1.54 + 4.75i)7-s + (−0.809 − 0.587i)9-s + (−1.61 − 1.17i)13-s + (−1.23 − 3.80i)15-s + (3.23 − 2.35i)17-s + (−0.927 + 2.85i)19-s + 5.00·21-s + 6·23-s + (3.39 − 10.4i)25-s + (−0.809 + 0.587i)27-s + (0.618 + 1.90i)29-s + (−0.809 − 0.587i)31-s + (16.1 + 11.7i)35-s + ⋯
L(s)  = 1  + (0.178 − 0.549i)3-s + (1.44 − 1.05i)5-s + (0.583 + 1.79i)7-s + (−0.269 − 0.195i)9-s + (−0.448 − 0.326i)13-s + (−0.319 − 0.982i)15-s + (0.784 − 0.570i)17-s + (−0.212 + 0.654i)19-s + 1.09·21-s + 1.25·23-s + (0.679 − 2.09i)25-s + (−0.155 + 0.113i)27-s + (0.114 + 0.353i)29-s + (−0.145 − 0.105i)31-s + (2.73 + 1.98i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1452\)    =    \(2^{2} \cdot 3 \cdot 11^{2}\)
Sign: $0.859 + 0.511i$
Analytic conductor: \(11.5942\)
Root analytic conductor: \(3.40503\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1452} (565, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1452,\ (\ :1/2),\ 0.859 + 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.533747806\)
\(L(\frac12)\) \(\approx\) \(2.533747806\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 \)
good5 \( 1 + (-3.23 + 2.35i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-1.54 - 4.75i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (1.61 + 1.17i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-3.23 + 2.35i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.927 - 2.85i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 + (-0.618 - 1.90i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.16 - 6.65i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-3.09 + 9.51i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (0.618 - 1.90i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.61 - 1.17i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.85 + 5.70i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-5.66 + 4.11i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 9T + 67T^{2} \)
71 \( 1 + (6.47 - 4.70i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.927 - 2.85i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-0.809 - 0.587i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (11.3 - 8.22i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 4T + 89T^{2} \)
97 \( 1 + (10.5 + 7.64i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.420206640278754483028408439069, −8.588933492697115959526292371800, −8.221323075610953316565136632296, −6.94162579795283379689461167867, −5.84472463692697281833932350122, −5.45160345556488225733483450551, −4.83457762558353667077275052659, −2.91881961919952244436162770559, −2.14012936376370668080989378736, −1.25602868507139935141121574193, 1.29925679032822798841239088972, 2.52335913116446855657830655543, 3.52405394058276227290993292552, 4.51768148810344788925841053543, 5.38398846443240431933170034666, 6.45413288893194315695370387564, 7.10198356045838381197637032450, 7.81701979536388702035591205272, 9.034990638428481958289299745808, 9.850714693488644154665867641067

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