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Properties

Label 2-21e2-147.104-c1-0-18
Degree $2$
Conductor $441$
Sign $-0.178 + 0.983i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.178 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.178 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.178 + 0.983i$
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: \((2,\ 441,\ (\ :1/2),\ -0.178 + 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.579553 - 0.694128i\)
\(L(\frac12)\) \(\approx\) \(0.579553 - 0.694128i\)
\(L(\frac{3}{2})\) 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

−10.89473608860250544468357574624, −10.05703368189612914887541930128, −8.870487959553769604549149032970, −7.890774340946594300826574998431, −7.13478158845652097934146243546, −5.91869862061738263764605592265, −5.21889529946379500134246289164, −4.03269018713005425239363209168, −3.19976028383418483210585034330, −0.42321009671574194715966742832, 2.58641187877169047751737187557, 3.30924084454124873008457282414, 4.34934158604073729361202504647, 5.29305637593344296590752740243, 6.92961819766143962990347513425, 7.50661559026732186672995410109, 8.561030757393776854117917181557, 9.886767776473897193894718169966, 10.78439516403549483166273448715, 11.44881903901637756354588419870

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