Properties

Label 2-21e2-441.142-c1-0-5
Degree $2$
Conductor $441$
Sign $-0.323 + 0.946i$
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  + (−0.416 + 1.06i)2-s + (−0.936 + 1.45i)3-s + (0.513 + 0.476i)4-s + (−1.98 + 0.954i)5-s + (−1.15 − 1.59i)6-s + (−1.83 + 1.90i)7-s + (−2.77 + 1.33i)8-s + (−1.24 − 2.72i)9-s + (−0.187 − 2.49i)10-s + (1.60 + 2.00i)11-s + (−1.17 + 0.302i)12-s + (1.37 − 3.51i)13-s + (−1.25 − 2.74i)14-s + (0.464 − 3.78i)15-s + (−0.157 − 2.10i)16-s + (1.50 − 1.39i)17-s + ⋯
L(s)  = 1  + (−0.294 + 0.750i)2-s + (−0.540 + 0.841i)3-s + (0.256 + 0.238i)4-s + (−0.886 + 0.426i)5-s + (−0.472 − 0.653i)6-s + (−0.694 + 0.719i)7-s + (−0.980 + 0.472i)8-s + (−0.415 − 0.909i)9-s + (−0.0592 − 0.790i)10-s + (0.482 + 0.605i)11-s + (−0.339 + 0.0873i)12-s + (0.382 − 0.974i)13-s + (−0.335 − 0.732i)14-s + (0.119 − 0.976i)15-s + (−0.0393 − 0.525i)16-s + (0.363 − 0.337i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.323 + 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.323 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.323 + 0.946i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.323 + 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.269260 - 0.376581i\)
\(L(\frac12)\) \(\approx\) \(0.269260 - 0.376581i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.936 - 1.45i)T \)
7 \( 1 + (1.83 - 1.90i)T \)
good2 \( 1 + (0.416 - 1.06i)T + (-1.46 - 1.36i)T^{2} \)
5 \( 1 + (1.98 - 0.954i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (-1.60 - 2.00i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (-1.37 + 3.51i)T + (-9.52 - 8.84i)T^{2} \)
17 \( 1 + (-1.50 + 1.39i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (-1.19 + 2.07i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.03 - 8.89i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (4.41 + 1.36i)T + (23.9 + 16.3i)T^{2} \)
31 \( 1 + (-1.61 + 2.79i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.05 - 1.55i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (4.74 + 3.23i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (0.447 - 0.305i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (-2.06 + 5.27i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (7.50 - 2.31i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (8.58 - 5.85i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (5.50 - 5.10i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (7.35 - 12.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.89 + 8.31i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-4.24 - 0.640i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (-1.58 - 2.74i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.331 + 0.844i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (-3.62 - 9.23i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (6.73 - 11.6i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.80751295338068023811262227555, −10.96289944285178828570126988098, −9.721412632834687811817006023483, −9.135725249320804449394278082866, −7.932491701318698387362873882538, −7.18193265067277704855652144856, −6.11267283486264597465491369855, −5.40411819143747868489549481424, −3.78563400287650540178086605414, −3.02251728023006791502100989605, 0.34263840634781087731271782263, 1.57037628054918599465307707416, 3.23628385063468389699584892351, 4.40048582938438533891846493103, 6.13037895266475504109725291014, 6.57161395073349734474432659124, 7.73855732432478978497520746431, 8.694062902990743915520688795481, 9.792237324306296568115636042055, 10.83706126285251096067564688909

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