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Properties

Label 2-21e2-441.47-c1-0-30
Degree $2$
Conductor $441$
Sign $-0.184 - 0.982i$
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  + (1.73 + 1.86i)2-s + (1.41 − 0.992i)3-s + (−0.335 + 4.47i)4-s + (1.23 + 1.54i)5-s + (4.31 + 0.931i)6-s + (−2.58 + 0.581i)7-s + (−4.96 + 3.95i)8-s + (1.03 − 2.81i)9-s + (−0.752 + 4.99i)10-s + (3.41 − 0.779i)11-s + (3.96 + 6.69i)12-s + (−3.09 − 3.33i)13-s + (−5.55 − 3.81i)14-s + (3.28 + 0.972i)15-s + (−7.11 − 1.07i)16-s + (−0.0470 − 0.628i)17-s + ⋯
L(s)  = 1  + (1.22 + 1.32i)2-s + (0.819 − 0.572i)3-s + (−0.167 + 2.23i)4-s + (0.552 + 0.692i)5-s + (1.76 + 0.380i)6-s + (−0.975 + 0.219i)7-s + (−1.75 + 1.39i)8-s + (0.343 − 0.939i)9-s + (−0.237 + 1.57i)10-s + (1.02 − 0.234i)11-s + (1.14 + 1.93i)12-s + (−0.858 − 0.924i)13-s + (−1.48 − 1.01i)14-s + (0.849 + 0.251i)15-s + (−1.77 − 0.268i)16-s + (−0.0114 − 0.152i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.184 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.184 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.184 - 0.982i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.184 - 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.05952 + 2.48117i\)
\(L(\frac12)\) \(\approx\) \(2.05952 + 2.48117i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.41 + 0.992i)T \)
7 \( 1 + (2.58 - 0.581i)T \)
good2 \( 1 + (-1.73 - 1.86i)T + (-0.149 + 1.99i)T^{2} \)
5 \( 1 + (-1.23 - 1.54i)T + (-1.11 + 4.87i)T^{2} \)
11 \( 1 + (-3.41 + 0.779i)T + (9.91 - 4.77i)T^{2} \)
13 \( 1 + (3.09 + 3.33i)T + (-0.971 + 12.9i)T^{2} \)
17 \( 1 + (0.0470 + 0.628i)T + (-16.8 + 2.53i)T^{2} \)
19 \( 1 + (5.31 + 3.06i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.622 - 1.29i)T + (-14.3 + 17.9i)T^{2} \)
29 \( 1 + (-1.94 - 2.86i)T + (-10.5 + 26.9i)T^{2} \)
31 \( 1 + (-7.04 - 4.06i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.85 - 2.63i)T + (13.5 - 34.4i)T^{2} \)
41 \( 1 + (-0.825 + 2.10i)T + (-30.0 - 27.8i)T^{2} \)
43 \( 1 + (-1.99 - 5.07i)T + (-31.5 + 29.2i)T^{2} \)
47 \( 1 + (1.17 - 1.08i)T + (3.51 - 46.8i)T^{2} \)
53 \( 1 + (1.66 - 2.43i)T + (-19.3 - 49.3i)T^{2} \)
59 \( 1 + (3.39 + 8.63i)T + (-43.2 + 40.1i)T^{2} \)
61 \( 1 + (8.47 - 0.635i)T + (60.3 - 9.09i)T^{2} \)
67 \( 1 + (-5.53 + 9.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.74 + 7.77i)T + (-44.2 + 55.5i)T^{2} \)
73 \( 1 + (-4.09 - 13.2i)T + (-60.3 + 41.1i)T^{2} \)
79 \( 1 + (4.93 + 8.55i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (10.1 + 9.41i)T + (6.20 + 82.7i)T^{2} \)
89 \( 1 + (12.0 + 11.1i)T + (6.65 + 88.7i)T^{2} \)
97 \( 1 + (-9.01 - 5.20i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.98997529201682811681910692900, −10.32369321365057138437307856394, −9.229702071278657407841162305005, −8.359674752304111113640583318335, −7.27377642245784093011044588918, −6.52094715020333235487687014739, −6.18445726792385958542367646592, −4.69852421336345980278929046161, −3.36568168737194536964073540636, −2.69648974996477948618109231734, 1.71093395550082201315206141260, 2.73923262152528362876286115910, 4.06984436151311093083216897464, 4.41167985617553674807230779883, 5.73187846393301243528344311199, 6.81864074053617100874495370362, 8.642251822786795428496152272101, 9.613596508739293305456640747151, 9.867104667128580729518937440190, 10.86901848675775641545166688477

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