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Properties

Label 2-21e2-63.59-c1-0-2
Degree $2$
Conductor $441$
Sign $0.987 - 0.156i$
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  + (−1.02 − 0.592i)2-s + (−1.25 − 1.19i)3-s + (−0.296 − 0.514i)4-s − 2.83·5-s + (0.576 + 1.97i)6-s + 3.07i·8-s + (0.136 + 2.99i)9-s + (2.91 + 1.68i)10-s + 0.157i·11-s + (−0.243 + 0.998i)12-s + (−3.41 − 1.97i)13-s + (3.55 + 3.39i)15-s + (1.23 − 2.13i)16-s + (−2.07 + 3.58i)17-s + (1.63 − 3.15i)18-s + (5.48 − 3.16i)19-s + ⋯
L(s)  = 1  + (−0.726 − 0.419i)2-s + (−0.723 − 0.690i)3-s + (−0.148 − 0.257i)4-s − 1.26·5-s + (0.235 + 0.804i)6-s + 1.08i·8-s + (0.0455 + 0.998i)9-s + (0.921 + 0.532i)10-s + 0.0475i·11-s + (−0.0702 + 0.288i)12-s + (−0.947 − 0.546i)13-s + (0.917 + 0.876i)15-s + (0.307 − 0.532i)16-s + (−0.502 + 0.870i)17-s + (0.385 − 0.744i)18-s + (1.25 − 0.726i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.156i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.987 - 0.156i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.987 - 0.156i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.298996 + 0.0234791i\)
\(L(\frac12)\) \(\approx\) \(0.298996 + 0.0234791i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.25 + 1.19i)T \)
7 \( 1 \)
good2 \( 1 + (1.02 + 0.592i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + 2.83T + 5T^{2} \)
11 \( 1 - 0.157iT - 11T^{2} \)
13 \( 1 + (3.41 + 1.97i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.07 - 3.58i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.48 + 3.16i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 0.546iT - 23T^{2} \)
29 \( 1 + (-4.02 + 2.32i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.112 - 0.0647i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.23 - 2.13i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.99 - 3.45i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.28 - 5.68i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.33 - 7.50i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.25 - 1.30i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.80 - 3.12i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.91 - 1.68i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.663 + 1.14i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.409iT - 71T^{2} \)
73 \( 1 + (-13.0 - 7.50i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.16 - 3.74i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.22 - 5.58i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.52 + 4.37i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.18 - 1.26i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.22472859351423439555722748803, −10.40603708486125912045750460912, −9.463112645136038624308901880638, −8.175071857860226855987319810361, −7.76199442593227044810505944943, −6.64522325400064296701168640703, −5.36801070339128860466182450415, −4.47764091714427317090532090815, −2.65771569895128498899395570511, −0.969816809736396376530597703921, 0.35689978904116261846871101471, 3.36918922139458222412340023546, 4.25196067970121287270524660055, 5.23342433081570863811631091239, 6.81883578769851407502646917510, 7.38124988140446106063461660240, 8.385198323364300988135801803999, 9.337861404862593640951600542315, 9.984121666344127608171509601910, 11.10958719893633047752066486146

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