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Properties

Label 2-21e2-63.20-c1-0-3
Degree $2$
Conductor $441$
Sign $-0.607 - 0.794i$
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  + (−1.02 − 0.589i)2-s + (−1.61 + 0.613i)3-s + (−0.305 − 0.529i)4-s + (2.16 + 3.75i)5-s + (2.01 + 0.327i)6-s + 3.07i·8-s + (2.24 − 1.98i)9-s − 5.10i·10-s + (−1.87 − 1.08i)11-s + (0.820 + 0.670i)12-s + (−2.25 + 1.30i)13-s + (−5.81 − 4.74i)15-s + (1.20 − 2.08i)16-s + 1.17·17-s + (−3.46 + 0.706i)18-s − 2.41i·19-s + ⋯
L(s)  = 1  + (−0.721 − 0.416i)2-s + (−0.935 + 0.354i)3-s + (−0.152 − 0.264i)4-s + (0.968 + 1.67i)5-s + (0.822 + 0.133i)6-s + 1.08i·8-s + (0.748 − 0.662i)9-s − 1.61i·10-s + (−0.564 − 0.325i)11-s + (0.236 + 0.193i)12-s + (−0.624 + 0.360i)13-s + (−1.50 − 1.22i)15-s + (0.300 − 0.520i)16-s + 0.284·17-s + (−0.816 + 0.166i)18-s − 0.554i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.607 - 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.607 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.607 - 0.794i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.607 - 0.794i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.185902 + 0.375941i\)
\(L(\frac12)\) \(\approx\) \(0.185902 + 0.375941i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.61 - 0.613i)T \)
7 \( 1 \)
good2 \( 1 + (1.02 + 0.589i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-2.16 - 3.75i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.87 + 1.08i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.25 - 1.30i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.17T + 17T^{2} \)
19 \( 1 + 2.41iT - 19T^{2} \)
23 \( 1 + (3.16 - 1.82i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.589 - 0.340i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.67 - 3.27i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.10T + 37T^{2} \)
41 \( 1 + (-3.68 - 6.38i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.12 - 3.68i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.57 - 6.18i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 3.23iT - 53T^{2} \)
59 \( 1 + (2.91 + 5.05i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.21 + 3.58i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.32 + 5.76i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.95iT - 71T^{2} \)
73 \( 1 - 11.9iT - 73T^{2} \)
79 \( 1 + (-4.87 + 8.44i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.796 - 1.37i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 6.09T + 89T^{2} \)
97 \( 1 + (-2.36 - 1.36i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.97818168696407517242432806901, −10.61574978491794123058389796237, −9.799918549191486896787863487618, −9.338063699042990006121038635569, −7.69155207293469336788256970661, −6.64865259863049282879868559845, −5.84290985083926007239756215451, −4.96748717561383373140558516470, −3.15226249398618251392206077386, −1.85568522071616026360154391116, 0.36881423864450969108686488998, 1.84199974954492287517378194081, 4.28013924828747856457747471987, 5.25664331604104529070311720106, 5.96992684796799146999196384982, 7.27888305965404393696258941105, 8.075603867860192679169848623723, 8.957026005033486927570672866965, 9.866425531884167193207769946103, 10.39182712665075638994284053913

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