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Properties

Label 2-21e2-63.47-c1-0-26
Degree $2$
Conductor $441$
Sign $0.551 + 0.834i$
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  + (0.850 − 0.490i)2-s + (0.831 − 1.51i)3-s + (−0.518 + 0.897i)4-s + 1.88·5-s + (−0.0387 − 1.69i)6-s + 2.98i·8-s + (−1.61 − 2.52i)9-s + (1.59 − 0.923i)10-s − 4.08i·11-s + (0.932 + 1.53i)12-s + (3.51 − 2.02i)13-s + (1.56 − 2.85i)15-s + (0.426 + 0.738i)16-s + (−0.810 − 1.40i)17-s + (−2.61 − 1.35i)18-s + (7.03 + 4.06i)19-s + ⋯
L(s)  = 1  + (0.601 − 0.347i)2-s + (0.480 − 0.877i)3-s + (−0.259 + 0.448i)4-s + 0.841·5-s + (−0.0158 − 0.693i)6-s + 1.05i·8-s + (−0.538 − 0.842i)9-s + (0.505 − 0.291i)10-s − 1.23i·11-s + (0.269 + 0.442i)12-s + (0.974 − 0.562i)13-s + (0.403 − 0.737i)15-s + (0.106 + 0.184i)16-s + (−0.196 − 0.340i)17-s + (−0.616 − 0.319i)18-s + (1.61 + 0.932i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.551 + 0.834i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.551 + 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.551 + 0.834i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (362, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.551 + 0.834i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.03582 - 1.09431i\)
\(L(\frac12)\) \(\approx\) \(2.03582 - 1.09431i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.831 + 1.51i)T \)
7 \( 1 \)
good2 \( 1 + (-0.850 + 0.490i)T + (1 - 1.73i)T^{2} \)
5 \( 1 - 1.88T + 5T^{2} \)
11 \( 1 + 4.08iT - 11T^{2} \)
13 \( 1 + (-3.51 + 2.02i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.810 + 1.40i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-7.03 - 4.06i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.31iT - 23T^{2} \)
29 \( 1 + (0.542 + 0.313i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.69 + 2.13i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.97 - 6.87i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.912 - 1.57i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.53 - 6.12i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.96 + 6.87i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7.24 + 4.18i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.08 - 7.07i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.24 - 1.87i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.26 - 10.8i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 14.4iT - 71T^{2} \)
73 \( 1 + (3.28 - 1.89i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.18 + 7.24i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.38 + 7.59i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.90 + 8.49i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-11.4 - 6.61i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.50229743893527218108215226098, −10.04713040545366513164407604198, −8.977912694862273886676088491442, −8.271296065060280496823793428287, −7.42081175800595803205532796503, −5.97046130961259085984182390716, −5.48102814923813891081282795839, −3.60446250697944973465305278699, −2.99697157519976821512952755606, −1.47727125602626826826344433948, 1.91236628657813558648849062962, 3.55389073008156912006950598426, 4.61202959881458650922947323880, 5.33922311576846215344693580208, 6.34545376170732689225287963517, 7.42754281265410669766685952963, 9.010940019320310084607679682440, 9.369849666299037782286370943727, 10.23533157122914300416070067507, 10.97265144709734026166196220906

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