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Properties

Label 2-21e2-9.4-c1-0-20
Degree $2$
Conductor $441$
Sign $0.0288 - 0.999i$
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  + (0.849 + 1.47i)2-s + (1.40 + 1.01i)3-s + (−0.444 + 0.769i)4-s + (0.474 − 0.822i)5-s + (−0.298 + 2.92i)6-s + 1.88·8-s + (0.944 + 2.84i)9-s + 1.61·10-s + (0.294 + 0.509i)11-s + (−1.40 + 0.630i)12-s + (2.50 − 4.34i)13-s + (1.5 − 0.673i)15-s + (2.49 + 4.31i)16-s − 7.58·17-s + (−3.38 + 3.80i)18-s − 4.46·19-s + ⋯
L(s)  = 1  + (0.600 + 1.04i)2-s + (0.810 + 0.585i)3-s + (−0.222 + 0.384i)4-s + (0.212 − 0.367i)5-s + (−0.121 + 1.19i)6-s + 0.667·8-s + (0.314 + 0.949i)9-s + 0.510·10-s + (0.0886 + 0.153i)11-s + (−0.405 + 0.181i)12-s + (0.696 − 1.20i)13-s + (0.387 − 0.173i)15-s + (0.623 + 1.07i)16-s − 1.83·17-s + (−0.798 + 0.897i)18-s − 1.02·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0288 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0288 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.0288 - 0.999i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.0288 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.88117 + 1.82770i\)
\(L(\frac12)\) \(\approx\) \(1.88117 + 1.82770i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.40 - 1.01i)T \)
7 \( 1 \)
good2 \( 1 + (-0.849 - 1.47i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-0.474 + 0.822i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.294 - 0.509i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.50 + 4.34i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 7.58T + 17T^{2} \)
19 \( 1 + 4.46T + 19T^{2} \)
23 \( 1 + (1.23 - 2.14i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.73 + 4.74i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.03 + 5.26i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6.98T + 37T^{2} \)
41 \( 1 + (-0.527 + 0.913i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.49 + 6.05i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.73 - 6.47i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 6.92T + 53T^{2} \)
59 \( 1 + (5.21 - 9.03i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.82 + 10.0i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.93 + 10.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.30T + 71T^{2} \)
73 \( 1 - 4.46T + 73T^{2} \)
79 \( 1 + (-0.666 - 1.15i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.84 - 4.92i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 0.843T + 89T^{2} \)
97 \( 1 + (-1.70 - 2.94i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.03310211328795037886318249317, −10.47791437939772699227109182238, −9.295118677009369133055186988307, −8.473428109807841640510810761698, −7.70646605603517782677983312799, −6.58978147520494244471581309308, −5.59570906925995599957857926182, −4.63751120343907968743462165057, −3.76958434207555256179874491494, −2.09657997584594553175666708995, 1.73247818190320437464972236007, 2.54231234591303676565738142132, 3.74432087027573890907676051741, 4.57131479472039949936238294841, 6.49095653929960724491983003177, 6.94793623135095394565966458231, 8.450333299914789715701010760402, 8.980863558060034753280704407157, 10.30817151692344022321541600858, 11.02296205784810525208230625548

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