Properties

Label 2-21e2-441.142-c1-0-37
Degree $2$
Conductor $441$
Sign $0.930 + 0.365i$
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  + (0.295 − 0.754i)2-s + (−0.878 + 1.49i)3-s + (0.985 + 0.913i)4-s + (2.57 − 1.23i)5-s + (0.865 + 1.10i)6-s + (0.114 − 2.64i)7-s + (2.44 − 1.17i)8-s + (−1.45 − 2.62i)9-s + (−0.172 − 2.30i)10-s + (0.206 + 0.259i)11-s + (−2.22 + 0.667i)12-s + (0.950 − 2.42i)13-s + (−1.95 − 0.868i)14-s + (−0.410 + 4.92i)15-s + (0.0368 + 0.491i)16-s + (−3.33 + 3.09i)17-s + ⋯
L(s)  = 1  + (0.209 − 0.533i)2-s + (−0.507 + 0.861i)3-s + (0.492 + 0.456i)4-s + (1.14 − 0.553i)5-s + (0.353 + 0.450i)6-s + (0.0432 − 0.999i)7-s + (0.862 − 0.415i)8-s + (−0.485 − 0.874i)9-s + (−0.0546 − 0.729i)10-s + (0.0623 + 0.0782i)11-s + (−0.643 + 0.192i)12-s + (0.263 − 0.671i)13-s + (−0.523 − 0.232i)14-s + (−0.106 + 1.27i)15-s + (0.00921 + 0.122i)16-s + (−0.808 + 0.750i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 + 0.365i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.930 + 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.930 + 0.365i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.930 + 0.365i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.81207 - 0.342877i\)
\(L(\frac12)\) \(\approx\) \(1.81207 - 0.342877i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.878 - 1.49i)T \)
7 \( 1 + (-0.114 + 2.64i)T \)
good2 \( 1 + (-0.295 + 0.754i)T + (-1.46 - 1.36i)T^{2} \)
5 \( 1 + (-2.57 + 1.23i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (-0.206 - 0.259i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (-0.950 + 2.42i)T + (-9.52 - 8.84i)T^{2} \)
17 \( 1 + (3.33 - 3.09i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (-2.60 + 4.51i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.899 - 3.93i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (-1.02 - 0.315i)T + (23.9 + 16.3i)T^{2} \)
31 \( 1 + (3.59 - 6.23i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-8.91 - 2.75i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (9.68 + 6.60i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (-1.50 + 1.02i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (4.30 - 10.9i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (-2.07 + 0.641i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (4.52 - 3.08i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (-8.35 + 7.74i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (7.21 - 12.5i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.790 - 3.46i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (10.6 + 1.60i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (-0.971 - 1.68i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.574 + 1.46i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (2.36 + 6.03i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (-3.72 + 6.45i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.96543718668313890899768645845, −10.34240257191311787472890826988, −9.598784942617956047040345947884, −8.571713798871557825138706092329, −7.22892450183213332213936037197, −6.22972297299722313451186446418, −5.13444077171120160327653255937, −4.16890200551048815356722675947, −3.07254756131099423931641105627, −1.39829677150683676807453817241, 1.78479265977573991241636541068, 2.51305325253212691647455362069, 4.92929030091608790216441526149, 5.92706365588227219768259023069, 6.25396624275319198350684052289, 7.11107913516897022195979046685, 8.223933484172963562252943613437, 9.450116062660930292780176692851, 10.34744383942865767729202855837, 11.37346509393026609948245080226

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