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Properties

Label 2-21e2-147.104-c1-0-7
Degree $2$
Conductor $441$
Sign $0.988 - 0.148i$
Analytic cond. $3.52140$
Root an. cond. $1.87654$
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  + (0.203 + 0.162i)2-s + (−0.430 − 1.88i)4-s + (2.46 + 1.18i)5-s + (−1.53 + 2.15i)7-s + (0.443 − 0.920i)8-s + (0.308 + 0.639i)10-s + (2.00 + 1.60i)11-s + (4.63 + 3.69i)13-s + (−0.660 + 0.189i)14-s + (−3.24 + 1.56i)16-s + (1.48 − 6.52i)17-s − 0.418i·19-s + (1.17 − 5.14i)20-s + (0.148 + 0.650i)22-s + (5.32 − 1.21i)23-s + ⋯
L(s)  = 1  + (0.143 + 0.114i)2-s + (−0.215 − 0.942i)4-s + (1.10 + 0.530i)5-s + (−0.579 + 0.815i)7-s + (0.156 − 0.325i)8-s + (0.0974 + 0.202i)10-s + (0.605 + 0.483i)11-s + (1.28 + 1.02i)13-s + (−0.176 + 0.0507i)14-s + (−0.810 + 0.390i)16-s + (0.361 − 1.58i)17-s − 0.0960i·19-s + (0.262 − 1.15i)20-s + (0.0316 + 0.138i)22-s + (1.11 − 0.253i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.148i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.988 - 0.148i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.988 - 0.148i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.72439 + 0.128954i\)
\(L(\frac12)\) \(\approx\) \(1.72439 + 0.128954i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (1.53 - 2.15i)T \)
good2 \( 1 + (-0.203 - 0.162i)T + (0.445 + 1.94i)T^{2} \)
5 \( 1 + (-2.46 - 1.18i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (-2.00 - 1.60i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (-4.63 - 3.69i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (-1.48 + 6.52i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 + 0.418iT - 19T^{2} \)
23 \( 1 + (-5.32 + 1.21i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (2.32 + 0.531i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 + 5.40iT - 31T^{2} \)
37 \( 1 + (1.37 - 6.03i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (-5.97 - 2.87i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (7.06 - 3.40i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (2.90 - 3.64i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-13.5 + 3.08i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (5.52 - 2.66i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (9.77 + 2.23i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + 8.37T + 67T^{2} \)
71 \( 1 + (6.34 - 1.44i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (5.32 - 4.24i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 - 0.883T + 79T^{2} \)
83 \( 1 + (6.46 + 8.10i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-1.58 - 1.98i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + 3.88iT - 97T^{2} \)
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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

−11.13198136319636861316982370737, −9.991944245012457319577530385693, −9.422343187023518042279328678800, −8.897555063505896140801937128391, −6.99177290799170366661297178358, −6.33494134880009193340268152100, −5.66189967358560576080468440228, −4.51161620971633200272262045552, −2.85191400258414437259141764685, −1.55182221147634277934025304096, 1.36157508625710973368347579944, 3.25742735433823309747813576153, 3.95074699101476476847892312479, 5.46021227869821623664204769067, 6.29103463438420893083753324524, 7.46354052078310146142926080246, 8.598655841537250171282260322969, 9.080749552602473430146798744657, 10.34259556428107292681143795817, 10.91649856864332726005994894373

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