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Properties

Label 2-21e2-63.47-c1-0-6
Degree $2$
Conductor $441$
Sign $-0.994 + 0.100i$
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  + (−0.575 + 0.332i)2-s + (0.537 + 1.64i)3-s + (−0.779 + 1.34i)4-s − 0.0283·5-s + (−0.855 − 0.768i)6-s − 2.36i·8-s + (−2.42 + 1.76i)9-s + (0.0162 − 0.00940i)10-s + 1.02i·11-s + (−2.64 − 0.558i)12-s + (−4.87 + 2.81i)13-s + (−0.0152 − 0.0466i)15-s + (−0.773 − 1.33i)16-s + (2.83 + 4.91i)17-s + (0.806 − 1.82i)18-s + (1.81 + 1.04i)19-s + ⋯
L(s)  = 1  + (−0.406 + 0.234i)2-s + (0.310 + 0.950i)3-s + (−0.389 + 0.674i)4-s − 0.0126·5-s + (−0.349 − 0.313i)6-s − 0.835i·8-s + (−0.807 + 0.589i)9-s + (0.00514 − 0.00297i)10-s + 0.308i·11-s + (−0.762 − 0.161i)12-s + (−1.35 + 0.781i)13-s + (−0.00392 − 0.0120i)15-s + (−0.193 − 0.334i)16-s + (0.688 + 1.19i)17-s + (0.190 − 0.429i)18-s + (0.415 + 0.240i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.994 + 0.100i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (362, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.994 + 0.100i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0364334 - 0.721074i\)
\(L(\frac12)\) \(\approx\) \(0.0364334 - 0.721074i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.537 - 1.64i)T \)
7 \( 1 \)
good2 \( 1 + (0.575 - 0.332i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + 0.0283T + 5T^{2} \)
11 \( 1 - 1.02iT - 11T^{2} \)
13 \( 1 + (4.87 - 2.81i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.83 - 4.91i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.81 - 1.04i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 7.26iT - 23T^{2} \)
29 \( 1 + (3.52 + 2.03i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.87 + 1.65i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.23 + 2.14i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.52 - 6.11i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.15 - 2.00i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.43 - 9.42i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-10.0 + 5.79i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.01 - 5.21i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.05 - 1.18i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.38 - 11.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.93iT - 71T^{2} \)
73 \( 1 + (9.43 - 5.44i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.80 - 13.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.07 + 5.32i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.02 + 10.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.77 - 3.91i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.57931783240863339914187174709, −10.29509339632734755005954376614, −9.716103727454375711252444608645, −8.966778331566879810396311581776, −8.027968583747302474424294453349, −7.32422035192700810181711317166, −5.88265254373029457364834858837, −4.53513511372640735993893249338, −3.90383075872770784603209253740, −2.51253437018836827497129515230, 0.48959768322058400964317030313, 1.99046177611202503076826296056, 3.27121149655519332784747746807, 5.18107529539850768587093911541, 5.75807398817679234993074473919, 7.31448618494403192682257949733, 7.73995444589508718981625950812, 9.060728035389090667462056505459, 9.558090962783520046553690579555, 10.57137867295340712256102936768

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