Properties

Label 2-21e2-441.122-c1-0-50
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: $\chi_{441} (122, \cdot )$
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

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

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