Properties

Label 2-21e2-63.58-c1-0-10
Degree $2$
Conductor $441$
Sign $0.698 - 0.715i$
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  − 1.84·2-s + (−1.39 + 1.02i)3-s + 1.39·4-s + (0.667 + 1.15i)5-s + (2.56 − 1.89i)6-s + 1.12·8-s + (0.880 − 2.86i)9-s + (−1.22 − 2.12i)10-s + (−0.756 + 1.31i)11-s + (−1.93 + 1.43i)12-s + (2.58 − 4.48i)13-s + (−2.11 − 0.923i)15-s − 4.84·16-s + (−0.774 − 1.34i)17-s + (−1.62 + 5.28i)18-s + (1.25 − 2.16i)19-s + ⋯
L(s)  = 1  − 1.30·2-s + (−0.804 + 0.594i)3-s + 0.695·4-s + (0.298 + 0.516i)5-s + (1.04 − 0.773i)6-s + 0.396·8-s + (0.293 − 0.955i)9-s + (−0.388 − 0.673i)10-s + (−0.228 + 0.395i)11-s + (−0.558 + 0.413i)12-s + (0.717 − 1.24i)13-s + (−0.547 − 0.238i)15-s − 1.21·16-s + (−0.187 − 0.325i)17-s + (−0.382 + 1.24i)18-s + (0.287 − 0.497i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.698 - 0.715i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.698 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.698 - 0.715i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.698 - 0.715i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.515667 + 0.217146i\)
\(L(\frac12)\) \(\approx\) \(0.515667 + 0.217146i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.39 - 1.02i)T \)
7 \( 1 \)
good2 \( 1 + 1.84T + 2T^{2} \)
5 \( 1 + (-0.667 - 1.15i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.756 - 1.31i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.58 + 4.48i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.774 + 1.34i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.25 + 2.16i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.68 - 6.37i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.0309 + 0.0536i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.84T + 31T^{2} \)
37 \( 1 + (0.281 - 0.487i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.51 - 7.81i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.09 - 8.83i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 9.51T + 47T^{2} \)
53 \( 1 + (-0.755 - 1.30i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 8.44T + 59T^{2} \)
61 \( 1 + 3.23T + 61T^{2} \)
67 \( 1 - 6.93T + 67T^{2} \)
71 \( 1 + 12.3T + 71T^{2} \)
73 \( 1 + (-1.37 - 2.38i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 5.91T + 79T^{2} \)
83 \( 1 + (2.80 + 4.85i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.703 - 1.21i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.09 - 10.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.93779960400879654437187340218, −10.22461564224855303268010190325, −9.650546842971226773045867162369, −8.733153672100647656500387155255, −7.64470817573030108043440872128, −6.74930008128797076166164546736, −5.65417577727565063870235479776, −4.56550204661676794427826111982, −2.98271630836785874010859626770, −0.989287337910654548576680380373, 0.860738025715233225981353269194, 2.01816875309205544487189948943, 4.31531864093039345581475280958, 5.48782731380195108105405114505, 6.60049782850361767073289305150, 7.35830747782597642568545626902, 8.581781776701065119319876529064, 8.922627912527083727206809710753, 10.20735942737701295104454032174, 10.81866498459777319582351199857

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