Properties

Label 2-21e2-441.5-c1-0-1
Degree $2$
Conductor $441$
Sign $-0.101 + 0.994i$
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  + (−1.08 + 0.869i)2-s + (0.851 + 1.50i)3-s + (−0.0127 + 0.0559i)4-s + (−2.47 − 1.68i)5-s + (−2.23 − 0.903i)6-s + (−0.888 + 2.49i)7-s + (−1.24 − 2.58i)8-s + (−1.54 + 2.56i)9-s + (4.16 − 0.311i)10-s + (−0.0467 − 0.309i)11-s + (−0.0952 + 0.0283i)12-s + (−0.257 − 1.70i)13-s + (−1.19 − 3.48i)14-s + (0.437 − 5.16i)15-s + (3.49 + 1.68i)16-s + (0.837 − 0.777i)17-s + ⋯
L(s)  = 1  + (−0.770 + 0.614i)2-s + (0.491 + 0.870i)3-s + (−0.00638 + 0.0279i)4-s + (−1.10 − 0.754i)5-s + (−0.913 − 0.368i)6-s + (−0.335 + 0.941i)7-s + (−0.439 − 0.913i)8-s + (−0.516 + 0.856i)9-s + (1.31 − 0.0986i)10-s + (−0.0140 − 0.0934i)11-s + (−0.0274 + 0.00819i)12-s + (−0.0714 − 0.473i)13-s + (−0.320 − 0.932i)14-s + (0.112 − 1.33i)15-s + (0.874 + 0.421i)16-s + (0.203 − 0.188i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.101 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.101 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.101 + 0.994i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.101 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0398197 - 0.0440989i\)
\(L(\frac12)\) \(\approx\) \(0.0398197 - 0.0440989i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.851 - 1.50i)T \)
7 \( 1 + (0.888 - 2.49i)T \)
good2 \( 1 + (1.08 - 0.869i)T + (0.445 - 1.94i)T^{2} \)
5 \( 1 + (2.47 + 1.68i)T + (1.82 + 4.65i)T^{2} \)
11 \( 1 + (0.0467 + 0.309i)T + (-10.5 + 3.24i)T^{2} \)
13 \( 1 + (0.257 + 1.70i)T + (-12.4 + 3.83i)T^{2} \)
17 \( 1 + (-0.837 + 0.777i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (4.78 + 2.76i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.766 + 2.48i)T + (-19.0 + 12.9i)T^{2} \)
29 \( 1 + (1.75 + 1.89i)T + (-2.16 + 28.9i)T^{2} \)
31 \( 1 - 2.95iT - 31T^{2} \)
37 \( 1 + (-0.479 - 0.147i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (0.230 - 3.07i)T + (-40.5 - 6.11i)T^{2} \)
43 \( 1 + (-0.255 - 3.40i)T + (-42.5 + 6.40i)T^{2} \)
47 \( 1 + (5.61 + 7.04i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (-3.24 - 10.5i)T + (-43.7 + 29.8i)T^{2} \)
59 \( 1 + (11.8 + 5.69i)T + (36.7 + 46.1i)T^{2} \)
61 \( 1 + (-6.30 + 1.43i)T + (54.9 - 26.4i)T^{2} \)
67 \( 1 + 13.3T + 67T^{2} \)
71 \( 1 + (4.11 + 0.938i)T + (63.9 + 30.8i)T^{2} \)
73 \( 1 + (-0.0526 + 0.348i)T + (-69.7 - 21.5i)T^{2} \)
79 \( 1 - 10.4T + 79T^{2} \)
83 \( 1 + (15.5 + 2.34i)T + (79.3 + 24.4i)T^{2} \)
89 \( 1 + (-6.22 - 15.8i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (5.61 - 3.24i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.79752880462254380640501609450, −10.65308724955608152187832445828, −9.578539688820692116835584413923, −8.853779038968285838467379456582, −8.347156375185083921589530237768, −7.66312633128438140628421024946, −6.30373050649529280398202048971, −4.97613667446633483393630927486, −3.97790593103252845606068050033, −2.87362949825628953874079497404, 0.04440349626722668164921517132, 1.69738486503525623623041520547, 3.10090353709442185031958922913, 4.09674606384814967320973789132, 6.04879557827515918599320627246, 7.07175216405261089815160806821, 7.76655967518506582349164781992, 8.580340580780255712647264640850, 9.629953277755445657962893627521, 10.52641300909354369494617094676

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