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Properties

Label 2-21e2-49.39-c1-0-10
Degree $2$
Conductor $441$
Sign $0.592 - 0.805i$
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  + (2.14 + 0.323i)2-s + (2.58 + 0.797i)4-s + (0.108 + 1.44i)5-s + (−1.12 + 2.39i)7-s + (1.38 + 0.665i)8-s + (−0.235 + 3.14i)10-s + (2.13 + 5.43i)11-s + (4.19 − 5.26i)13-s + (−3.19 + 4.77i)14-s + (−1.72 − 1.17i)16-s + (−0.330 − 0.306i)17-s + (−3.04 − 5.27i)19-s + (−0.874 + 3.83i)20-s + (2.81 + 12.3i)22-s + (1.11 − 1.03i)23-s + ⋯
L(s)  = 1  + (1.51 + 0.228i)2-s + (1.29 + 0.398i)4-s + (0.0485 + 0.647i)5-s + (−0.426 + 0.904i)7-s + (0.488 + 0.235i)8-s + (−0.0744 + 0.993i)10-s + (0.642 + 1.63i)11-s + (1.16 − 1.45i)13-s + (−0.853 + 1.27i)14-s + (−0.431 − 0.293i)16-s + (−0.0802 − 0.0744i)17-s + (−0.699 − 1.21i)19-s + (−0.195 + 0.857i)20-s + (0.600 + 2.63i)22-s + (0.231 − 0.215i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.592 - 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.592 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.592 - 0.805i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.592 - 0.805i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.68218 + 1.35595i\)
\(L(\frac12)\) \(\approx\) \(2.68218 + 1.35595i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.12 - 2.39i)T \)
good2 \( 1 + (-2.14 - 0.323i)T + (1.91 + 0.589i)T^{2} \)
5 \( 1 + (-0.108 - 1.44i)T + (-4.94 + 0.745i)T^{2} \)
11 \( 1 + (-2.13 - 5.43i)T + (-8.06 + 7.48i)T^{2} \)
13 \( 1 + (-4.19 + 5.26i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 + (0.330 + 0.306i)T + (1.27 + 16.9i)T^{2} \)
19 \( 1 + (3.04 + 5.27i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.11 + 1.03i)T + (1.71 - 22.9i)T^{2} \)
29 \( 1 + (-0.284 + 1.24i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + (-1.54 + 2.67i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.71 - 0.530i)T + (30.5 - 20.8i)T^{2} \)
41 \( 1 + (4.75 + 2.29i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (6.67 - 3.21i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (3.94 + 0.594i)T + (44.9 + 13.8i)T^{2} \)
53 \( 1 + (-7.69 - 2.37i)T + (43.7 + 29.8i)T^{2} \)
59 \( 1 + (0.530 - 7.08i)T + (-58.3 - 8.79i)T^{2} \)
61 \( 1 + (-0.0365 + 0.0112i)T + (50.4 - 34.3i)T^{2} \)
67 \( 1 + (-0.443 + 0.767i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.582 + 2.55i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-5.85 + 0.881i)T + (69.7 - 21.5i)T^{2} \)
79 \( 1 + (-8.35 - 14.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.47 - 1.84i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-0.735 + 1.87i)T + (-65.2 - 60.5i)T^{2} \)
97 \( 1 + 5.13T + 97T^{2} \)
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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

−11.55813586004877148612166093822, −10.59436670058293600016525049695, −9.531694616661940181176835988270, −8.453731530482318856358518613364, −6.96956820465435386452254995697, −6.48799163537186798962998991267, −5.48300284984324123637103041481, −4.51676212650183767503550851791, −3.32939853512109993507054952085, −2.42078815808348351736738735029, 1.46546320139895328869306480961, 3.49265920824331080810570005366, 3.88621534105870906036591689743, 5.03899867101999059734422831336, 6.25849325871486532661351406708, 6.62342574740409582039308785927, 8.410837768940549278524039794757, 9.036191962618974236994785604822, 10.49344757382550532887038861423, 11.29092904143026621803973135602

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