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Properties

Label 2-21e2-441.184-c1-0-17
Degree $2$
Conductor $441$
Sign $-0.870 - 0.491i$
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  + (1.97 + 0.950i)2-s + (−0.880 + 1.49i)3-s + (1.74 + 2.18i)4-s + (−1.72 + 1.59i)5-s + (−3.15 + 2.10i)6-s + (0.702 + 2.55i)7-s + (0.386 + 1.69i)8-s + (−1.45 − 2.62i)9-s + (−4.91 + 1.51i)10-s + (1.75 − 1.19i)11-s + (−4.79 + 0.675i)12-s + (0.407 − 0.278i)13-s + (−1.03 + 5.69i)14-s + (−0.868 − 3.97i)15-s + (0.396 − 1.73i)16-s + (−1.06 − 0.160i)17-s + ⋯
L(s)  = 1  + (1.39 + 0.671i)2-s + (−0.508 + 0.861i)3-s + (0.871 + 1.09i)4-s + (−0.770 + 0.715i)5-s + (−1.28 + 0.859i)6-s + (0.265 + 0.964i)7-s + (0.136 + 0.599i)8-s + (−0.483 − 0.875i)9-s + (−1.55 + 0.479i)10-s + (0.528 − 0.360i)11-s + (−1.38 + 0.195i)12-s + (0.113 − 0.0771i)13-s + (−0.277 + 1.52i)14-s + (−0.224 − 1.02i)15-s + (0.0990 − 0.434i)16-s + (−0.257 − 0.0388i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.870 - 0.491i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.870 - 0.491i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.870 - 0.491i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (184, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.870 - 0.491i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.540698 + 2.05628i\)
\(L(\frac12)\) \(\approx\) \(0.540698 + 2.05628i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.880 - 1.49i)T \)
7 \( 1 + (-0.702 - 2.55i)T \)
good2 \( 1 + (-1.97 - 0.950i)T + (1.24 + 1.56i)T^{2} \)
5 \( 1 + (1.72 - 1.59i)T + (0.373 - 4.98i)T^{2} \)
11 \( 1 + (-1.75 + 1.19i)T + (4.01 - 10.2i)T^{2} \)
13 \( 1 + (-0.407 + 0.278i)T + (4.74 - 12.1i)T^{2} \)
17 \( 1 + (1.06 + 0.160i)T + (16.2 + 5.01i)T^{2} \)
19 \( 1 + (1.55 - 2.69i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.66 - 4.23i)T + (-16.8 - 15.6i)T^{2} \)
29 \( 1 + (-0.161 - 0.0242i)T + (27.7 + 8.54i)T^{2} \)
31 \( 1 - 6.34T + 31T^{2} \)
37 \( 1 + (-2.77 - 7.05i)T + (-27.1 + 25.1i)T^{2} \)
41 \( 1 + (-1.49 - 0.460i)T + (33.8 + 23.0i)T^{2} \)
43 \( 1 + (-9.57 + 2.95i)T + (35.5 - 24.2i)T^{2} \)
47 \( 1 + (-4.11 - 1.98i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (-3.75 + 9.57i)T + (-38.8 - 36.0i)T^{2} \)
59 \( 1 + (-1.01 + 4.44i)T + (-53.1 - 25.5i)T^{2} \)
61 \( 1 + (0.814 - 1.02i)T + (-13.5 - 59.4i)T^{2} \)
67 \( 1 + 9.89T + 67T^{2} \)
71 \( 1 + (5.48 + 6.88i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (3.61 + 2.46i)T + (26.6 + 67.9i)T^{2} \)
79 \( 1 - 6.83T + 79T^{2} \)
83 \( 1 + (10.0 + 6.88i)T + (30.3 + 77.2i)T^{2} \)
89 \( 1 + (-1.07 + 14.3i)T + (-88.0 - 13.2i)T^{2} \)
97 \( 1 + (-7.39 - 12.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.79426058274328166566665820747, −10.96898527370416054197048989035, −9.798095491055550514465501452122, −8.664088489137337160094216585128, −7.53283600330650493236128427031, −6.34037056834918376533450734550, −5.83140184792538626927344955028, −4.74366043244825895987859237868, −3.85931965505328408057929912900, −2.99894283354203569281888575388, 0.983239562013856939713789719078, 2.48640901093338324904017932426, 4.26271559799901346332801159311, 4.45120819484843818021873908841, 5.80704113191990972727002166516, 6.80080846200469769754290228775, 7.78141564348214379010721533504, 8.771231175679652087542805263314, 10.47293629944541689574374602707, 11.12962195368221513924229995330

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