Properties

Label 2-21e2-9.7-c1-0-25
Degree $2$
Conductor $441$
Sign $0.533 + 0.845i$
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  + (1.19 − 2.06i)2-s + (1.34 + 1.08i)3-s + (−1.84 − 3.20i)4-s + (1.46 + 2.52i)5-s + (3.85 − 1.49i)6-s − 4.05·8-s + (0.637 + 2.93i)9-s + 6.97·10-s + (0.676 − 1.17i)11-s + (0.987 − 6.32i)12-s + (−0.733 − 1.26i)13-s + (−0.779 + 4.99i)15-s + (−1.13 + 1.96i)16-s − 3.31·17-s + (6.82 + 2.18i)18-s + 2.20·19-s + ⋯
L(s)  = 1  + (0.843 − 1.46i)2-s + (0.778 + 0.627i)3-s + (−0.924 − 1.60i)4-s + (0.653 + 1.13i)5-s + (1.57 − 0.608i)6-s − 1.43·8-s + (0.212 + 0.977i)9-s + 2.20·10-s + (0.204 − 0.353i)11-s + (0.284 − 1.82i)12-s + (−0.203 − 0.352i)13-s + (−0.201 + 1.29i)15-s + (−0.284 + 0.492i)16-s − 0.802·17-s + (1.60 + 0.514i)18-s + 0.506·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.533 + 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.533 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.533 + 0.845i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.533 + 0.845i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.46112 - 1.35679i\)
\(L(\frac12)\) \(\approx\) \(2.46112 - 1.35679i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.34 - 1.08i)T \)
7 \( 1 \)
good2 \( 1 + (-1.19 + 2.06i)T + (-1 - 1.73i)T^{2} \)
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 \( 1 + 3.31T + 17T^{2} \)
19 \( 1 - 2.20T + 19T^{2} \)
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 \( 1 + (1.63 + 2.83i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 10.8T + 37T^{2} \)
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 \( 1 + (1.98 - 3.44i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 6.45T + 53T^{2} \)
59 \( 1 + (-6.10 - 10.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.279 - 0.484i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.40 + 11.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.9T + 71T^{2} \)
73 \( 1 + 10.4T + 73T^{2} \)
79 \( 1 + (0.383 - 0.664i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.983 - 1.70i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 6.40T + 89T^{2} \)
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

−10.85271525755819690604871292999, −10.31272382724822596043325709878, −9.667915885137344199638800108454, −8.654881484304614714604805266238, −7.21700438775901506193914524493, −5.89152174502276492377736251844, −4.77977937570827080987610021662, −3.67054182822774135195028996306, −2.86145403290015530706922731350, −2.01915495759851157004178041024, 1.79144653416798600015110309099, 3.63293664451329669973882684955, 4.76613075754981307508919051029, 5.59665394305336976432095297699, 6.71417034522240047383998360799, 7.31412799055537705283872153772, 8.502985882327170149331792932979, 8.895000824456389354143385935309, 9.930028942644466127170472916282, 11.83151480104775796097877538778

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