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Properties

Label 2-21e2-63.5-c1-0-19
Degree $2$
Conductor $441$
Sign $0.111 + 0.993i$
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  − 2.09i·2-s + (1.72 + 0.109i)3-s − 2.39·4-s + (1.04 + 1.80i)5-s + (0.228 − 3.62i)6-s + 0.819i·8-s + (2.97 + 0.377i)9-s + (3.79 − 2.18i)10-s + (2.79 + 1.61i)11-s + (−4.13 − 0.260i)12-s + (2.68 + 1.55i)13-s + (1.60 + 3.24i)15-s − 3.06·16-s + (−0.816 − 1.41i)17-s + (0.790 − 6.23i)18-s + (−4.79 − 2.76i)19-s + ⋯
L(s)  = 1  − 1.48i·2-s + (0.998 + 0.0629i)3-s − 1.19·4-s + (0.467 + 0.809i)5-s + (0.0933 − 1.47i)6-s + 0.289i·8-s + (0.992 + 0.125i)9-s + (1.19 − 0.692i)10-s + (0.843 + 0.486i)11-s + (−1.19 − 0.0753i)12-s + (0.745 + 0.430i)13-s + (0.415 + 0.837i)15-s − 0.766·16-s + (−0.197 − 0.342i)17-s + (0.186 − 1.46i)18-s + (−1.09 − 0.634i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.111 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.111 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.111 + 0.993i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.111 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56195 - 1.39633i\)
\(L(\frac12)\) \(\approx\) \(1.56195 - 1.39633i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.72 - 0.109i)T \)
7 \( 1 \)
good2 \( 1 + 2.09iT - 2T^{2} \)
5 \( 1 + (-1.04 - 1.80i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.79 - 1.61i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.68 - 1.55i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.816 + 1.41i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.79 + 2.76i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.00 - 0.580i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (7.05 - 4.07i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 5.96iT - 31T^{2} \)
37 \( 1 + (-2.82 + 4.89i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.35 - 2.34i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.974 + 1.68i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 8.13T + 47T^{2} \)
53 \( 1 + (5.27 - 3.04i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 3.96T + 59T^{2} \)
61 \( 1 - 4.79iT - 61T^{2} \)
67 \( 1 + 0.673T + 67T^{2} \)
71 \( 1 - 7.01iT - 71T^{2} \)
73 \( 1 + (-2.96 + 1.71i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 14.1T + 79T^{2} \)
83 \( 1 + (-1.54 - 2.67i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.45 - 4.25i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.07 + 1.20i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.93999838545933277003898871512, −10.03986816396388839637514682947, −9.362406985070661656129363540579, −8.685509706212492882911894031824, −7.21398262025074679411249329535, −6.39223386871476899227064795362, −4.42208605692561492991608791089, −3.64501999721411538706970235182, −2.53143192101435435987631866293, −1.70298781048515246146949504575, 1.70764027146353197247708481128, 3.63017630275556704851826796779, 4.73409249637292926464658960796, 5.94616102127889801246132076490, 6.61514103417723423741075505743, 7.82961675989962643346709444550, 8.544805666339025213111907374859, 8.969524129458617095055101257062, 9.983622768546429312143855246076, 11.29863604300289092918538646032

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