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Properties

Label 2-21e2-49.16-c1-0-18
Degree $2$
Conductor $441$
Sign $-0.203 + 0.979i$
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  + (−0.293 − 0.748i)2-s + (0.992 − 0.920i)4-s + (1.11 − 0.762i)5-s + (1.80 − 1.93i)7-s + (−2.42 − 1.16i)8-s + (−0.899 − 0.613i)10-s + (−1.27 − 0.192i)11-s + (−0.117 + 0.146i)13-s + (−1.97 − 0.779i)14-s + (0.0402 − 0.537i)16-s + (2.86 − 0.882i)17-s + (−2.34 + 4.06i)19-s + (0.407 − 1.78i)20-s + (0.230 + 1.01i)22-s + (2.09 + 0.645i)23-s + ⋯
L(s)  = 1  + (−0.207 − 0.529i)2-s + (0.496 − 0.460i)4-s + (0.500 − 0.340i)5-s + (0.680 − 0.732i)7-s + (−0.858 − 0.413i)8-s + (−0.284 − 0.193i)10-s + (−0.384 − 0.0579i)11-s + (−0.0324 + 0.0406i)13-s + (−0.529 − 0.208i)14-s + (0.0100 − 0.134i)16-s + (0.693 − 0.213i)17-s + (−0.538 + 0.933i)19-s + (0.0911 − 0.399i)20-s + (0.0491 + 0.215i)22-s + (0.436 + 0.134i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.203 + 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.203 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.203 + 0.979i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.203 + 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.974291 - 1.19772i\)
\(L(\frac12)\) \(\approx\) \(0.974291 - 1.19772i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-1.80 + 1.93i)T \)
good2 \( 1 + (0.293 + 0.748i)T + (-1.46 + 1.36i)T^{2} \)
5 \( 1 + (-1.11 + 0.762i)T + (1.82 - 4.65i)T^{2} \)
11 \( 1 + (1.27 + 0.192i)T + (10.5 + 3.24i)T^{2} \)
13 \( 1 + (0.117 - 0.146i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 + (-2.86 + 0.882i)T + (14.0 - 9.57i)T^{2} \)
19 \( 1 + (2.34 - 4.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.09 - 0.645i)T + (19.0 + 12.9i)T^{2} \)
29 \( 1 + (-1.11 + 4.90i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + (1.06 + 1.83i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.51 + 3.26i)T + (2.76 + 36.8i)T^{2} \)
41 \( 1 + (-9.47 - 4.56i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (-1.77 + 0.853i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (2.97 + 7.58i)T + (-34.4 + 31.9i)T^{2} \)
53 \( 1 + (5.78 - 5.36i)T + (3.96 - 52.8i)T^{2} \)
59 \( 1 + (2.53 + 1.72i)T + (21.5 + 54.9i)T^{2} \)
61 \( 1 + (-8.23 - 7.63i)T + (4.55 + 60.8i)T^{2} \)
67 \( 1 + (-6.34 - 10.9i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.310 - 1.36i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-4.84 + 12.3i)T + (-53.5 - 49.6i)T^{2} \)
79 \( 1 + (6.97 - 12.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-10.7 - 13.4i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (2.26 - 0.340i)T + (85.0 - 26.2i)T^{2} \)
97 \( 1 - 14.5T + 97T^{2} \)
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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

−10.82075334029174205047690733924, −10.07931819898960667366760549035, −9.374159655948208635476303960174, −8.140151013648605799136275623026, −7.23307757967842470471056459856, −6.02528810908208256240508682372, −5.19653412102265011224729290518, −3.79664250255725538142832999772, −2.28367100491352576213334221806, −1.11656022178125932404626893470, 2.10955918974275699381091098338, 3.10904924906454323671637366482, 4.87244863737365877203818067395, 5.88695467377647010243861280519, 6.73893398287230521218991438535, 7.74215310129173948253263253493, 8.511137486098174134702871873692, 9.359138231484120497498257076027, 10.62615536171886864288591145838, 11.27322945613132213077503189129

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