Properties

Label 2-21e2-63.25-c1-0-31
Degree $2$
Conductor $441$
Sign $-0.632 + 0.774i$
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  + 1.29·2-s + (−1.47 − 0.910i)3-s − 0.310·4-s + (1.76 − 3.05i)5-s + (−1.91 − 1.18i)6-s − 3.00·8-s + (1.34 + 2.68i)9-s + (2.29 − 3.96i)10-s + (−0.589 − 1.02i)11-s + (0.457 + 0.282i)12-s + (−1.61 − 2.78i)13-s + (−5.37 + 2.89i)15-s − 3.28·16-s + (−2.45 + 4.24i)17-s + (1.74 + 3.48i)18-s + (−3.43 − 5.94i)19-s + ⋯
L(s)  = 1  + 0.919·2-s + (−0.850 − 0.525i)3-s − 0.155·4-s + (0.788 − 1.36i)5-s + (−0.782 − 0.482i)6-s − 1.06·8-s + (0.447 + 0.894i)9-s + (0.724 − 1.25i)10-s + (−0.177 − 0.307i)11-s + (0.132 + 0.0815i)12-s + (−0.446 − 0.773i)13-s + (−1.38 + 0.747i)15-s − 0.820·16-s + (−0.594 + 1.02i)17-s + (0.411 + 0.821i)18-s + (−0.787 − 1.36i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.632 + 0.774i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.632 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.632 + 0.774i$
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: \((2,\ 441,\ (\ :1/2),\ -0.632 + 0.774i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.552564 - 1.16475i\)
\(L(\frac12)\) \(\approx\) \(0.552564 - 1.16475i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.47 + 0.910i)T \)
7 \( 1 \)
good2 \( 1 - 1.29T + 2T^{2} \)
5 \( 1 + (-1.76 + 3.05i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.589 + 1.02i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.61 + 2.78i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.45 - 4.24i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.43 + 5.94i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.14 + 3.72i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.36 + 2.35i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.92T + 31T^{2} \)
37 \( 1 + (-4.88 - 8.45i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.32 - 5.76i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.83 + 8.37i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 0.633T + 47T^{2} \)
53 \( 1 + (-1.11 + 1.92i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 8.21T + 59T^{2} \)
61 \( 1 - 9.65T + 61T^{2} \)
67 \( 1 - 5.33T + 67T^{2} \)
71 \( 1 + 3.27T + 71T^{2} \)
73 \( 1 + (-0.519 + 0.898i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 1.00T + 79T^{2} \)
83 \( 1 + (-3.65 + 6.33i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.02 - 10.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.46 + 9.46i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.00431817251946235449075798158, −9.981175762498402177951142140423, −8.885085521052444666102868322111, −8.186043660796247497228678166341, −6.54660367373983432210755451142, −5.86235266714418270663698857609, −4.94794480776386618337422351153, −4.46296317928651628167172319452, −2.44944082091681709711641138686, −0.66668126496521616838844127576, 2.45852056486293399175756110031, 3.73704586603385102934923486149, 4.72319446171387560917011278913, 5.74864351491564182492500157286, 6.41336822450722673423507206133, 7.29676505421148290012965770180, 9.190928060969364411722760474393, 9.752119307217234187169979745237, 10.68814971996511134215321870003, 11.42450264588758571892002128027

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