Properties

Label 2-21e2-441.151-c1-0-10
Degree $2$
Conductor $441$
Sign $0.808 - 0.588i$
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  + (0.971 − 0.468i)2-s + (−1.40 − 1.01i)3-s + (−0.521 + 0.653i)4-s + (1.57 + 1.46i)5-s + (−1.83 − 0.329i)6-s + (−2.14 − 1.54i)7-s + (−0.680 + 2.98i)8-s + (0.939 + 2.84i)9-s + (2.21 + 0.683i)10-s + (1.27 + 0.866i)11-s + (1.39 − 0.388i)12-s + (5.49 + 3.74i)13-s + (−2.81 − 0.494i)14-s + (−0.727 − 3.65i)15-s + (0.362 + 1.58i)16-s + (−1.49 + 0.225i)17-s + ⋯
L(s)  = 1  + (0.687 − 0.330i)2-s + (−0.810 − 0.585i)3-s + (−0.260 + 0.326i)4-s + (0.704 + 0.653i)5-s + (−0.750 − 0.134i)6-s + (−0.812 − 0.583i)7-s + (−0.240 + 1.05i)8-s + (0.313 + 0.949i)9-s + (0.700 + 0.216i)10-s + (0.383 + 0.261i)11-s + (0.402 − 0.112i)12-s + (1.52 + 1.03i)13-s + (−0.751 − 0.132i)14-s + (−0.187 − 0.942i)15-s + (0.0905 + 0.396i)16-s + (−0.363 + 0.0547i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.808 - 0.588i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.808 - 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.808 - 0.588i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.808 - 0.588i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34581 + 0.438048i\)
\(L(\frac12)\) \(\approx\) \(1.34581 + 0.438048i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.14 + 1.54i)T \)
good2 \( 1 + (-0.971 + 0.468i)T + (1.24 - 1.56i)T^{2} \)
5 \( 1 + (-1.57 - 1.46i)T + (0.373 + 4.98i)T^{2} \)
11 \( 1 + (-1.27 - 0.866i)T + (4.01 + 10.2i)T^{2} \)
13 \( 1 + (-5.49 - 3.74i)T + (4.74 + 12.1i)T^{2} \)
17 \( 1 + (1.49 - 0.225i)T + (16.2 - 5.01i)T^{2} \)
19 \( 1 + (-0.0797 - 0.138i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.01 - 7.67i)T + (-16.8 + 15.6i)T^{2} \)
29 \( 1 + (0.550 - 0.0829i)T + (27.7 - 8.54i)T^{2} \)
31 \( 1 - 2.50T + 31T^{2} \)
37 \( 1 + (2.78 - 7.08i)T + (-27.1 - 25.1i)T^{2} \)
41 \( 1 + (-1.12 + 0.345i)T + (33.8 - 23.0i)T^{2} \)
43 \( 1 + (10.4 + 3.23i)T + (35.5 + 24.2i)T^{2} \)
47 \( 1 + (-10.6 + 5.11i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (1.33 + 3.39i)T + (-38.8 + 36.0i)T^{2} \)
59 \( 1 + (1.10 + 4.84i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (-2.37 - 2.97i)T + (-13.5 + 59.4i)T^{2} \)
67 \( 1 + 7.07T + 67T^{2} \)
71 \( 1 + (-2.11 + 2.65i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (0.118 - 0.0809i)T + (26.6 - 67.9i)T^{2} \)
79 \( 1 + 4.29T + 79T^{2} \)
83 \( 1 + (1.31 - 0.899i)T + (30.3 - 77.2i)T^{2} \)
89 \( 1 + (-0.741 - 9.89i)T + (-88.0 + 13.2i)T^{2} \)
97 \( 1 + (-3.34 + 5.79i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.39936345661207114739870247917, −10.60555098198691021000896815055, −9.585299575517815820522918663029, −8.488248297627153169530769689756, −7.09239086550543562676501584059, −6.48242812060218231907390164428, −5.59838232717340212157898216216, −4.29445441571280862093736798772, −3.27814190873949137617117757316, −1.73570058857306441037457225019, 0.868209041250510021083671451783, 3.32260697819391038211513240365, 4.43992729316121258734883815202, 5.48924183098591870742774766142, 5.98776469548670465967517592687, 6.66147284402553412111331638270, 8.758030475524420899106030459164, 9.185869231782509474209422471860, 10.20172813850975176933152226356, 10.87158805739990998145399239403

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