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Properties

Label 2-21e2-441.104-c1-0-26
Degree $2$
Conductor $441$
Sign $-0.277 + 0.960i$
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  + (0.244 − 1.62i)2-s + (−1.73 − 0.0837i)3-s + (−0.666 − 0.205i)4-s + (−0.532 + 0.363i)5-s + (−0.559 + 2.78i)6-s + (1.40 + 2.24i)7-s + (0.928 − 1.92i)8-s + (2.98 + 0.289i)9-s + (0.459 + 0.953i)10-s + (0.457 − 3.03i)11-s + (1.13 + 0.411i)12-s + (2.65 − 1.04i)13-s + (3.98 − 1.72i)14-s + (0.951 − 0.583i)15-s + (−4.05 − 2.76i)16-s + (−0.255 + 1.11i)17-s + ⋯
L(s)  = 1  + (0.173 − 1.14i)2-s + (−0.998 − 0.0483i)3-s + (−0.333 − 0.102i)4-s + (−0.238 + 0.162i)5-s + (−0.228 + 1.13i)6-s + (0.529 + 0.848i)7-s + (0.328 − 0.681i)8-s + (0.995 + 0.0966i)9-s + (0.145 + 0.301i)10-s + (0.137 − 0.914i)11-s + (0.327 + 0.118i)12-s + (0.735 − 0.288i)13-s + (1.06 − 0.461i)14-s + (0.245 − 0.150i)15-s + (−1.01 − 0.691i)16-s + (−0.0618 + 0.271i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.277 + 0.960i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.277 + 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.758345 - 1.00842i\)
\(L(\frac12)\) \(\approx\) \(0.758345 - 1.00842i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.73 + 0.0837i)T \)
7 \( 1 + (-1.40 - 2.24i)T \)
good2 \( 1 + (-0.244 + 1.62i)T + (-1.91 - 0.589i)T^{2} \)
5 \( 1 + (0.532 - 0.363i)T + (1.82 - 4.65i)T^{2} \)
11 \( 1 + (-0.457 + 3.03i)T + (-10.5 - 3.24i)T^{2} \)
13 \( 1 + (-2.65 + 1.04i)T + (9.52 - 8.84i)T^{2} \)
17 \( 1 + (0.255 - 1.11i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 + 2.78iT - 19T^{2} \)
23 \( 1 + (-2.56 + 8.32i)T + (-19.0 - 12.9i)T^{2} \)
29 \( 1 + (-0.370 - 1.19i)T + (-23.9 + 16.3i)T^{2} \)
31 \( 1 + (-4.77 - 2.75i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.37 + 10.4i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (3.15 - 2.15i)T + (14.9 - 38.1i)T^{2} \)
43 \( 1 + (-1.21 - 0.828i)T + (15.7 + 40.0i)T^{2} \)
47 \( 1 + (-4.72 - 0.712i)T + (44.9 + 13.8i)T^{2} \)
53 \( 1 + (-1.35 + 0.308i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (-0.720 + 9.61i)T + (-58.3 - 8.79i)T^{2} \)
61 \( 1 + (-0.916 - 2.97i)T + (-50.4 + 34.3i)T^{2} \)
67 \( 1 + (2.45 - 4.25i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.56 + 1.49i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (0.705 - 0.562i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + (-6.63 - 11.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.16 - 5.51i)T + (-60.8 - 56.4i)T^{2} \)
89 \( 1 + (4.79 + 6.00i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + (11.1 - 6.42i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.04979553793606190023250940689, −10.53820134894339883209446964441, −9.269410568715324531492227248082, −8.272904387747876162641868880184, −6.93790971450767195424182148239, −6.01828645576601418482912707477, −4.93320764781766458392937419820, −3.79633374109014225343733010161, −2.50402318138854887626695745872, −0.993671105277169647104068881662, 1.50445205738099838435560220474, 4.09214721058622847791471215972, 4.81924447203646434030283476366, 5.82783523196820875415010404343, 6.74345697483800302909433381355, 7.44644675653197961491539361077, 8.222292371632758703572008499541, 9.676759239410038025586346145527, 10.56651019469756420353788806508, 11.47952168112322645390579344536

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