Properties

Label 2-21e2-63.25-c1-0-7
Degree $2$
Conductor $441$
Sign $0.349 - 0.936i$
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  − 2.38·2-s + (1.61 − 0.624i)3-s + 3.69·4-s + (−1.46 + 2.52i)5-s + (−3.85 + 1.49i)6-s − 4.05·8-s + (2.22 − 2.01i)9-s + (3.48 − 6.03i)10-s + (0.676 + 1.17i)11-s + (5.97 − 2.30i)12-s + (0.733 + 1.26i)13-s + (−0.779 + 4.99i)15-s + 2.27·16-s + (−1.65 + 2.86i)17-s + (−5.29 + 4.81i)18-s + (1.10 + 1.91i)19-s + ⋯
L(s)  = 1  − 1.68·2-s + (0.932 − 0.360i)3-s + 1.84·4-s + (−0.653 + 1.13i)5-s + (−1.57 + 0.608i)6-s − 1.43·8-s + (0.740 − 0.672i)9-s + (1.10 − 1.90i)10-s + (0.204 + 0.353i)11-s + (1.72 − 0.666i)12-s + (0.203 + 0.352i)13-s + (−0.201 + 1.29i)15-s + 0.568·16-s + (−0.401 + 0.695i)17-s + (−1.24 + 1.13i)18-s + (0.253 + 0.438i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.349 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.349 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.349 - 0.936i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (214, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.349 - 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.590205 + 0.409646i\)
\(L(\frac12)\) \(\approx\) \(0.590205 + 0.409646i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.61 + 0.624i)T \)
7 \( 1 \)
good2 \( 1 + 2.38T + 2T^{2} \)
5 \( 1 + (1.46 - 2.52i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.676 - 1.17i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.733 - 1.26i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.65 - 2.86i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.10 - 1.91i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.31 - 2.27i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.521 + 0.903i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.27T + 31T^{2} \)
37 \( 1 + (-5.43 - 9.41i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.904 - 1.56i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.17 - 3.76i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.97T + 47T^{2} \)
53 \( 1 + (3.22 - 5.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 12.2T + 59T^{2} \)
61 \( 1 + 0.559T + 61T^{2} \)
67 \( 1 - 12.8T + 67T^{2} \)
71 \( 1 - 12.9T + 71T^{2} \)
73 \( 1 + (5.22 - 9.05i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 0.767T + 79T^{2} \)
83 \( 1 + (-0.983 + 1.70i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.20 + 5.54i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.14 + 7.17i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.10774043220486642879209460111, −10.06906226976758575271601179208, −9.522403760048596482591038695954, −8.410265507698815188657364936160, −7.88724089287398289202101497767, −7.02486868789906326309732300147, −6.43762968190600772976757207116, −3.98491163037614810955031097286, −2.81133418574209614392623897199, −1.59250992717026269522047426366, 0.73532301455088347258011576102, 2.30207758835454709085545040551, 3.82974495078955425086750790604, 5.08017706579305355521548410541, 6.84799277585032827273593252791, 7.82019908538109729896655705235, 8.392541674702479090148535933338, 9.076631533437187522615537591700, 9.616563994853592893918315416134, 10.70913517077312500084003370402

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