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Properties

Label 2-21e2-63.25-c1-0-4
Degree $2$
Conductor $441$
Sign $0.210 - 0.977i$
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  + 0.0683·2-s + (0.539 − 1.64i)3-s − 1.99·4-s + (−1.33 + 2.30i)5-s + (0.0368 − 0.112i)6-s − 0.273·8-s + (−2.41 − 1.77i)9-s + (−0.0910 + 0.157i)10-s + (0.799 + 1.38i)11-s + (−1.07 + 3.28i)12-s + (2.62 + 4.54i)13-s + (3.07 + 3.43i)15-s + 3.97·16-s + (−3.27 + 5.67i)17-s + (−0.165 − 0.121i)18-s + (0.950 + 1.64i)19-s + ⋯
L(s)  = 1  + 0.0483·2-s + (0.311 − 0.950i)3-s − 0.997·4-s + (−0.595 + 1.03i)5-s + (0.0150 − 0.0459i)6-s − 0.0965·8-s + (−0.805 − 0.592i)9-s + (−0.0287 + 0.0498i)10-s + (0.241 + 0.417i)11-s + (−0.310 + 0.948i)12-s + (0.728 + 1.26i)13-s + (0.794 + 0.887i)15-s + 0.992·16-s + (−0.793 + 1.37i)17-s + (−0.0389 − 0.0286i)18-s + (0.218 + 0.377i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.210 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.210 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.210 - 0.977i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (214, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.210 - 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.628937 + 0.508039i\)
\(L(\frac12)\) \(\approx\) \(0.628937 + 0.508039i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.539 + 1.64i)T \)
7 \( 1 \)
good2 \( 1 - 0.0683T + 2T^{2} \)
5 \( 1 + (1.33 - 2.30i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.799 - 1.38i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.62 - 4.54i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.27 - 5.67i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.950 - 1.64i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.53 + 2.65i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.19 - 5.53i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.71T + 31T^{2} \)
37 \( 1 + (2.11 + 3.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.69 - 6.40i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.63 + 9.75i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.79T + 47T^{2} \)
53 \( 1 + (4.44 - 7.70i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 10.8T + 59T^{2} \)
61 \( 1 + 2.71T + 61T^{2} \)
67 \( 1 + 3.32T + 67T^{2} \)
71 \( 1 + 12.3T + 71T^{2} \)
73 \( 1 + (-1.09 + 1.90i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 0.813T + 79T^{2} \)
83 \( 1 + (-3.41 + 5.92i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.235 - 0.407i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.57 - 4.46i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.30788707235450034061647283998, −10.60055429581464191646765496410, −9.152510205834621475629621732795, −8.684846651425723186124373896396, −7.55960388103875057443795522368, −6.80879493360649072580197951352, −5.87412494974582427843527881888, −4.19670261208735796485461190096, −3.43047494795860217928530582576, −1.76213050820702986217455398855, 0.51231658936477873940983561703, 3.17515185495591482732990282620, 4.12619088227743167500299527043, 4.97300523895405703935087645100, 5.69624036800570943536883276552, 7.66604680839253158686159075446, 8.478677117344005187688716040854, 9.088113064418234419217894541987, 9.729967632227811575394914855137, 10.94303180463335637778551701918

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