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Properties

Label 2-21e2-63.38-c1-0-15
Degree $2$
Conductor $441$
Sign $0.270 - 0.962i$
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  + 2.37i·2-s + (−1.65 + 0.523i)3-s − 3.62·4-s + (1.71 − 2.97i)5-s + (−1.24 − 3.91i)6-s − 3.86i·8-s + (2.45 − 1.72i)9-s + (7.05 + 4.07i)10-s + (−0.271 + 0.156i)11-s + (5.99 − 1.90i)12-s + (5.09 − 2.94i)13-s + (−1.27 + 5.81i)15-s + 1.91·16-s + (0.476 − 0.825i)17-s + (4.10 + 5.81i)18-s + (−1.09 + 0.630i)19-s + ⋯
L(s)  = 1  + 1.67i·2-s + (−0.953 + 0.302i)3-s − 1.81·4-s + (0.768 − 1.33i)5-s + (−0.507 − 1.59i)6-s − 1.36i·8-s + (0.817 − 0.576i)9-s + (2.23 + 1.28i)10-s + (−0.0819 + 0.0473i)11-s + (1.73 − 0.548i)12-s + (1.41 − 0.816i)13-s + (−0.329 + 1.50i)15-s + 0.479·16-s + (0.115 − 0.200i)17-s + (0.967 + 1.37i)18-s + (−0.250 + 0.144i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.270 - 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.270 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.270 - 0.962i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.270 - 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.885502 + 0.671305i\)
\(L(\frac12)\) \(\approx\) \(0.885502 + 0.671305i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.65 - 0.523i)T \)
7 \( 1 \)
good2 \( 1 - 2.37iT - 2T^{2} \)
5 \( 1 + (-1.71 + 2.97i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.271 - 0.156i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-5.09 + 2.94i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.476 + 0.825i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.09 - 0.630i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.91 - 3.41i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.43 - 1.98i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.23iT - 31T^{2} \)
37 \( 1 + (2.68 + 4.65i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.0699 + 0.121i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.44 + 2.49i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.01T + 47T^{2} \)
53 \( 1 + (-10.3 - 5.98i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 1.64T + 59T^{2} \)
61 \( 1 - 2.97iT - 61T^{2} \)
67 \( 1 + 1.86T + 67T^{2} \)
71 \( 1 - 10.9iT - 71T^{2} \)
73 \( 1 + (0.354 + 0.204i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 10.4T + 79T^{2} \)
83 \( 1 + (-4.00 + 6.92i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.05 + 1.83i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (10.5 + 6.06i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.24163741743942363903297185289, −10.16369823109740174138900781144, −9.130700902748215614171330709996, −8.626632868142791547935238203828, −7.47588598033951731826908322302, −6.36323615280573319387867778549, −5.58298493179335297449925653195, −5.18127702733206375358401955290, −4.05973923969640681080585967503, −0.984340536479439538553338699822, 1.33515304233748371978609646521, 2.49928107154000449223838246822, 3.67271277481054516351414997596, 4.95045737635725005139879390732, 6.31828073710108107775200996879, 6.83768112093577877541198646246, 8.566619373178823012314048673807, 9.661958093751551667677863325384, 10.58901953267738103236605266947, 10.81579095357429402629315989087

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