Properties

Label 2-21e2-63.16-c1-0-30
Degree $2$
Conductor $441$
Sign $0.701 + 0.712i$
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  + (0.439 + 0.761i)2-s + (1.11 − 1.32i)3-s + (0.613 − 1.06i)4-s − 1.34·5-s + (1.5 + 0.264i)6-s + 2.83·8-s + (−0.520 − 2.95i)9-s + (−0.592 − 1.02i)10-s + 1.65·11-s + (−0.726 − 1.99i)12-s + (−1.68 − 2.91i)13-s + (−1.5 + 1.78i)15-s + (0.0209 + 0.0362i)16-s + (0.233 + 0.405i)17-s + (2.02 − 1.69i)18-s + (−1.61 + 2.79i)19-s + ⋯
L(s)  = 1  + (0.310 + 0.538i)2-s + (0.642 − 0.766i)3-s + (0.306 − 0.531i)4-s − 0.602·5-s + (0.612 + 0.107i)6-s + 1.00·8-s + (−0.173 − 0.984i)9-s + (−0.187 − 0.324i)10-s + 0.498·11-s + (−0.209 − 0.576i)12-s + (−0.467 − 0.809i)13-s + (−0.387 + 0.461i)15-s + (0.00523 + 0.00906i)16-s + (0.0567 + 0.0982i)17-s + (0.476 − 0.399i)18-s + (−0.370 + 0.641i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.701 + 0.712i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.701 + 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.87184 - 0.784458i\)
\(L(\frac12)\) \(\approx\) \(1.87184 - 0.784458i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.11 + 1.32i)T \)
7 \( 1 \)
good2 \( 1 + (-0.439 - 0.761i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + 1.34T + 5T^{2} \)
11 \( 1 - 1.65T + 11T^{2} \)
13 \( 1 + (1.68 + 2.91i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.233 - 0.405i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.61 - 2.79i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 8.94T + 23T^{2} \)
29 \( 1 + (3.13 - 5.42i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.61 + 7.99i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.61 - 7.99i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.70 - 2.95i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.20 + 3.82i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.67 - 8.10i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.286 - 0.497i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.19 - 9.00i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.81 - 6.61i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.298 - 0.516i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.554T + 71T^{2} \)
73 \( 1 + (-1.02 - 1.77i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.20 - 2.08i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.52 - 13.0i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.54 + 7.86i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.949 - 1.64i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.11230668527169531739935341145, −10.07479174096784668167605742988, −9.023700141116652858582322621536, −7.942787735238470584668901909057, −7.32461559070774217535956468217, −6.46619058874958595268693337597, −5.47414082987462139635001727830, −4.14332904254177473246108783603, −2.79722891893787142943577745980, −1.23454846392654247227327934858, 2.16362808756966934270816126251, 3.30322834398127462867770409850, 4.14746444960558267365807583154, 4.97936377199446691309229423952, 6.84997138915573757726807970200, 7.60554651637607125938360216187, 8.665732836084758500486423050286, 9.375678763818626664947178045532, 10.56379653417027652622457286155, 11.25671461459845713030920012265

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