Properties

Label 2-21e2-147.104-c1-0-3
Degree $2$
Conductor $441$
Sign $0.987 - 0.159i$
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  + (−1.16 − 0.926i)2-s + (0.0461 + 0.202i)4-s + (2.75 + 1.32i)5-s + (−2.64 + 0.00862i)7-s + (−1.15 + 2.39i)8-s + (−1.97 − 4.09i)10-s + (4.38 + 3.49i)11-s + (−0.126 − 0.100i)13-s + (3.08 + 2.44i)14-s + (3.93 − 1.89i)16-s + (−0.351 + 1.53i)17-s + 5.64i·19-s + (−0.141 + 0.618i)20-s + (−1.85 − 8.12i)22-s + (−4.97 + 1.13i)23-s + ⋯
L(s)  = 1  + (−0.821 − 0.654i)2-s + (0.0230 + 0.101i)4-s + (1.23 + 0.594i)5-s + (−0.999 + 0.00325i)7-s + (−0.408 + 0.848i)8-s + (−0.624 − 1.29i)10-s + (1.32 + 1.05i)11-s + (−0.0350 − 0.0279i)13-s + (0.823 + 0.652i)14-s + (0.984 − 0.474i)16-s + (−0.0852 + 0.373i)17-s + 1.29i·19-s + (−0.0315 + 0.138i)20-s + (−0.395 − 1.73i)22-s + (−1.03 + 0.236i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.159i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.987 - 0.159i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.987 - 0.159i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.966813 + 0.0777087i\)
\(L(\frac12)\) \(\approx\) \(0.966813 + 0.0777087i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (2.64 - 0.00862i)T \)
good2 \( 1 + (1.16 + 0.926i)T + (0.445 + 1.94i)T^{2} \)
5 \( 1 + (-2.75 - 1.32i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (-4.38 - 3.49i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (0.126 + 0.100i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (0.351 - 1.53i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 - 5.64iT - 19T^{2} \)
23 \( 1 + (4.97 - 1.13i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (-3.45 - 0.788i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 + 0.856iT - 31T^{2} \)
37 \( 1 + (-0.520 + 2.28i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (-10.3 - 5.00i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (-10.9 + 5.26i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-6.32 + 7.92i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (6.39 - 1.45i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (8.14 - 3.92i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (0.616 + 0.140i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + 10.8T + 67T^{2} \)
71 \( 1 + (4.40 - 1.00i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (-5.53 + 4.41i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + 9.69T + 79T^{2} \)
83 \( 1 + (-4.61 - 5.78i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-1.71 - 2.14i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + 6.81iT - 97T^{2} \)
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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

−10.71421420606255911947158208657, −10.09983305280705370232323861921, −9.567546641287978108866707325855, −8.970150426112242884047566574791, −7.51068859593684537772134324157, −6.26049990356900638611248589293, −5.86824720232625761090965621122, −4.02281896425827372007996347369, −2.54426053097837690695157566357, −1.56389214938347481107439079598, 0.869756168758315076086778784472, 2.86476016817683317249192028470, 4.27041392045129055513220996668, 6.03705374843359501291930524911, 6.24948885553440506732036544521, 7.39983920088199155009811471573, 8.706668666361967111202281675787, 9.260318068315510151123521791405, 9.631358085312949748062024821856, 10.84240914370154153112747748710

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