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Properties

Label 2-1452-11.3-c1-0-3
Degree $2$
Conductor $1452$
Sign $-0.642 - 0.766i$
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 + (1.30 + 0.951i)5-s + (−0.0729 + 0.224i)7-s + (−0.809 + 0.587i)9-s + (−3.42 + 2.48i)13-s + (−0.499 + 1.53i)15-s + (−0.118 − 0.0857i)17-s + (2.11 + 6.51i)19-s − 0.236·21-s − 5·23-s + (−0.736 − 2.26i)25-s + (−0.809 − 0.587i)27-s + (−0.618 + 1.90i)29-s + (−2.73 + 1.98i)31-s + (−0.309 + 0.224i)35-s + ⋯
L(s)  = 1  + (0.178 + 0.549i)3-s + (0.585 + 0.425i)5-s + (−0.0275 + 0.0848i)7-s + (−0.269 + 0.195i)9-s + (−0.950 + 0.690i)13-s + (−0.129 + 0.397i)15-s + (−0.0286 − 0.0207i)17-s + (0.485 + 1.49i)19-s − 0.0515·21-s − 1.04·23-s + (−0.147 − 0.453i)25-s + (−0.155 − 0.113i)27-s + (−0.114 + 0.353i)29-s + (−0.491 + 0.357i)31-s + (−0.0522 + 0.0379i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1452\)    =    \(2^{2} \cdot 3 \cdot 11^{2}\)
Sign: $-0.642 - 0.766i$
Analytic conductor: \(11.5942\)
Root analytic conductor: \(3.40503\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1452} (1213, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1452,\ (\ :1/2),\ -0.642 - 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.445076340\)
\(L(\frac12)\) \(\approx\) \(1.445076340\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.30 - 0.951i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.0729 - 0.224i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (3.42 - 2.48i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.118 + 0.0857i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.11 - 6.51i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 5T + 23T^{2} \)
29 \( 1 + (0.618 - 1.90i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (2.73 - 1.98i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.690 + 2.12i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.69 - 8.28i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 2.52T + 43T^{2} \)
47 \( 1 + (-2.95 - 9.09i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (5.35 - 3.88i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-3.64 + 11.2i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (8.78 + 6.37i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 4.38T + 67T^{2} \)
71 \( 1 + (-11.7 - 8.55i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-4.85 + 14.9i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (7.04 - 5.11i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (1.42 + 1.03i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 5.18T + 89T^{2} \)
97 \( 1 + (12.1 - 8.83i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.646288149339882392226458694350, −9.387859327521079198361819125813, −8.150610992570081514884144667032, −7.51693694487059948937307424567, −6.39040437797865112624860267706, −5.76358290155086385297368987602, −4.73161280143911498120240878389, −3.84724000288795026297637033609, −2.75682038748005692918026590354, −1.78075936738433047008292557584, 0.53307402057853990245427053955, 1.97103444875439180676193588617, 2.83409420054908193705295798806, 4.12486898709802626723506992266, 5.27224120514055789025162436032, 5.78914277708794566295110981149, 6.99840902855503553477909143732, 7.49109296844971202969273047060, 8.463827620673337090485490331177, 9.246333128310640993111910801737

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