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Properties

Label 2-21e2-63.41-c1-0-1
Degree $2$
Conductor $441$
Sign $-0.778 + 0.627i$
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  + (−1.28 + 0.742i)2-s + (0.729 + 1.57i)3-s + (0.101 − 0.176i)4-s + (0.154 − 0.267i)5-s + (−2.10 − 1.47i)6-s − 2.66i·8-s + (−1.93 + 2.29i)9-s + 0.457i·10-s + (−2.73 + 1.58i)11-s + (0.351 + 0.0312i)12-s + (−3.00 − 1.73i)13-s + (0.532 + 0.0472i)15-s + (2.18 + 3.78i)16-s − 4.88·17-s + (0.784 − 4.38i)18-s + 5.34i·19-s + ⋯
L(s)  = 1  + (−0.909 + 0.524i)2-s + (0.421 + 0.906i)3-s + (0.0509 − 0.0882i)4-s + (0.0689 − 0.119i)5-s + (−0.859 − 0.603i)6-s − 0.942i·8-s + (−0.644 + 0.764i)9-s + 0.144i·10-s + (−0.825 + 0.476i)11-s + (0.101 + 0.00901i)12-s + (−0.833 − 0.481i)13-s + (0.137 + 0.0121i)15-s + (0.545 + 0.945i)16-s − 1.18·17-s + (0.184 − 1.03i)18-s + 1.22i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.778 + 0.627i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.778 + 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.778 + 0.627i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.778 + 0.627i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.124045 - 0.351670i\)
\(L(\frac12)\) \(\approx\) \(0.124045 - 0.351670i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.729 - 1.57i)T \)
7 \( 1 \)
good2 \( 1 + (1.28 - 0.742i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-0.154 + 0.267i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.73 - 1.58i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.00 + 1.73i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 4.88T + 17T^{2} \)
19 \( 1 - 5.34iT - 19T^{2} \)
23 \( 1 + (5.17 + 2.98i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.70 + 1.56i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (6.51 + 3.76i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 11.8T + 37T^{2} \)
41 \( 1 + (2.58 - 4.48i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.75 - 4.76i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.23 + 7.33i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 0.0855iT - 53T^{2} \)
59 \( 1 + (-1.04 + 1.80i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.69 - 2.71i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.0554 + 0.0959i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.78iT - 71T^{2} \)
73 \( 1 - 9.61iT - 73T^{2} \)
79 \( 1 + (2.56 + 4.44i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.42 - 7.66i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 1.87T + 89T^{2} \)
97 \( 1 + (-10.9 + 6.34i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.35685629347189472092191567955, −10.22477748404463702203674606117, −9.861554712798536461076985602681, −8.958759617598071374006322326693, −8.058174106749533604077012652802, −7.54233589749641154093162142731, −6.14241133505123365589334441194, −4.88070505008587825621868714113, −3.89278425767660078878967225103, −2.44230664936933527939176815562, 0.27960385592906596483902758524, 2.00633330941647734277829669954, 2.81957903576652545260950492244, 4.74460392144890981420245063575, 6.02387316572991947341874864577, 7.13387463395718121357781604330, 8.003541871697279238808721897236, 8.874374969357577773484809521052, 9.473040833594151882409386145749, 10.61673952183445404602888346210

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