Properties

Label 2-21e2-63.4-c1-0-25
Degree $2$
Conductor $441$
Sign $0.266 + 0.963i$
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  + (−0.673 + 1.16i)2-s + (0.592 − 1.62i)3-s + (0.0923 + 0.160i)4-s − 2.53·5-s + (1.5 + 1.78i)6-s − 2.94·8-s + (−2.29 − 1.92i)9-s + (1.70 − 2.95i)10-s + 0.467·11-s + (0.315 − 0.0555i)12-s + (2.91 − 5.04i)13-s + (−1.50 + 4.12i)15-s + (1.79 − 3.11i)16-s + (1.93 − 3.35i)17-s + (3.79 − 1.38i)18-s + (−1.09 − 1.89i)19-s + ⋯
L(s)  = 1  + (−0.476 + 0.825i)2-s + (0.342 − 0.939i)3-s + (0.0461 + 0.0800i)4-s − 1.13·5-s + (0.612 + 0.729i)6-s − 1.04·8-s + (−0.766 − 0.642i)9-s + (0.539 − 0.934i)10-s + 0.141·11-s + (0.0909 − 0.0160i)12-s + (0.807 − 1.39i)13-s + (−0.387 + 1.06i)15-s + (0.449 − 0.778i)16-s + (0.470 − 0.814i)17-s + (0.895 − 0.325i)18-s + (−0.250 − 0.434i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.266 + 0.963i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.266 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.561932 - 0.427492i\)
\(L(\frac12)\) \(\approx\) \(0.561932 - 0.427492i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.592 + 1.62i)T \)
7 \( 1 \)
good2 \( 1 + (0.673 - 1.16i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 2.53T + 5T^{2} \)
11 \( 1 - 0.467T + 11T^{2} \)
13 \( 1 + (-2.91 + 5.04i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.93 + 3.35i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.09 + 1.89i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 0.106T + 23T^{2} \)
29 \( 1 + (4.39 + 7.60i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.84 + 6.65i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.84 - 6.65i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.11 - 1.92i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.613 + 1.06i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.66 - 4.61i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.358 + 0.620i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.368 - 0.637i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.479 + 0.829i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.81 - 8.34i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.2T + 71T^{2} \)
73 \( 1 + (5.13 - 8.89i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.31 + 10.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.36 + 2.36i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.05 + 7.02i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.80 + 11.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.36160031678721719407023815205, −9.692863179355618876917439353876, −8.656040147994965025035214901757, −7.81647181645256048064788988755, −7.67341173532183029447612276759, −6.52639508468209451145082004480, −5.64009487632801313532923384073, −3.77321975279774710630389761234, −2.79217243020816888945773277591, −0.49314541248281505733290936560, 1.80164826984576420069899669777, 3.47539630486383945661475046234, 3.96567197639062972090645501305, 5.42782652800769948317404003880, 6.70009089693359759202765420727, 8.070841189413322385682450661076, 8.874339607897783300002280763248, 9.492367825196043435881099954166, 10.68213414886402672366338846371, 11.01213528553389218188084299506

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