Properties

Label 2-21e2-441.142-c1-0-31
Degree $2$
Conductor $441$
Sign $-0.0543 + 0.998i$
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  + (0.581 − 1.48i)2-s + (−1.51 − 0.844i)3-s + (−0.392 − 0.363i)4-s + (1.51 − 0.731i)5-s + (−2.13 + 1.75i)6-s + (2.03 + 1.68i)7-s + (2.10 − 1.01i)8-s + (1.57 + 2.55i)9-s + (−0.200 − 2.67i)10-s + (2.16 + 2.72i)11-s + (0.285 + 0.881i)12-s + (2.32 − 5.92i)13-s + (3.68 − 2.03i)14-s + (−2.91 − 0.176i)15-s + (−0.357 − 4.77i)16-s + (−5.44 + 5.05i)17-s + ⋯
L(s)  = 1  + (0.411 − 1.04i)2-s + (−0.873 − 0.487i)3-s + (−0.196 − 0.181i)4-s + (0.678 − 0.326i)5-s + (−0.870 + 0.714i)6-s + (0.770 + 0.637i)7-s + (0.743 − 0.357i)8-s + (0.524 + 0.851i)9-s + (−0.0634 − 0.846i)10-s + (0.654 + 0.820i)11-s + (0.0824 + 0.254i)12-s + (0.645 − 1.64i)13-s + (0.985 − 0.544i)14-s + (−0.752 − 0.0455i)15-s + (−0.0893 − 1.19i)16-s + (−1.32 + 1.22i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0543 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0543 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.0543 + 0.998i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.0543 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22880 - 1.29746i\)
\(L(\frac12)\) \(\approx\) \(1.22880 - 1.29746i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.51 + 0.844i)T \)
7 \( 1 + (-2.03 - 1.68i)T \)
good2 \( 1 + (-0.581 + 1.48i)T + (-1.46 - 1.36i)T^{2} \)
5 \( 1 + (-1.51 + 0.731i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (-2.16 - 2.72i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (-2.32 + 5.92i)T + (-9.52 - 8.84i)T^{2} \)
17 \( 1 + (5.44 - 5.05i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (-1.72 + 2.98i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.708 - 3.10i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (4.34 + 1.34i)T + (23.9 + 16.3i)T^{2} \)
31 \( 1 + (-1.08 + 1.88i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (9.21 + 2.84i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (-2.85 - 1.94i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (-1.50 + 1.02i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (-2.61 + 6.65i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (-0.992 + 0.306i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (3.12 - 2.12i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (2.96 - 2.75i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (-5.86 + 10.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.78 - 12.2i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (0.944 + 0.142i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (-5.60 - 9.71i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.08 + 10.4i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (-0.249 - 0.636i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (3.87 - 6.71i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.02789515344126582249372421084, −10.45967419748112412586673795145, −9.339898349874982696444475992895, −8.129894154397179418318410743230, −7.09212785414609815402250349410, −5.85581740285633804442847167780, −5.12394210092134874498633087261, −3.96839376951252708883612519940, −2.22522475645036610675950077261, −1.43611568271745002491784032176, 1.64216234102870109034369761632, 4.02293604548806507965710765363, 4.76100162565105640612167335336, 5.83985785028992957781616400077, 6.57934372428606119650762614226, 7.12727714164435470809137551256, 8.607388270218082852898636883212, 9.529076588582759264179467311091, 10.76050715750739734597926144657, 11.14922842664387044978350560204

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