Properties

Label 2-21e2-63.47-c1-0-14
Degree $2$
Conductor $441$
Sign $0.732 - 0.681i$
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  + (−1.80 + 1.04i)2-s + (1.73 + 0.0239i)3-s + (1.17 − 2.03i)4-s − 3.30·5-s + (−3.15 + 1.76i)6-s + 0.717i·8-s + (2.99 + 0.0828i)9-s + (5.96 − 3.44i)10-s − 2.66i·11-s + (2.07 − 3.48i)12-s + (2.11 − 1.21i)13-s + (−5.72 − 0.0790i)15-s + (1.59 + 2.76i)16-s + (3.59 + 6.21i)17-s + (−5.49 + 2.97i)18-s + (4.24 + 2.45i)19-s + ⋯
L(s)  = 1  + (−1.27 + 0.736i)2-s + (0.999 + 0.0138i)3-s + (0.586 − 1.01i)4-s − 1.47·5-s + (−1.28 + 0.719i)6-s + 0.253i·8-s + (0.999 + 0.0276i)9-s + (1.88 − 1.08i)10-s − 0.802i·11-s + (0.600 − 1.00i)12-s + (0.585 − 0.338i)13-s + (−1.47 − 0.0203i)15-s + (0.399 + 0.691i)16-s + (0.870 + 1.50i)17-s + (−1.29 + 0.701i)18-s + (0.974 + 0.562i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.732 - 0.681i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.732 - 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.732 - 0.681i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (362, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.732 - 0.681i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.789528 + 0.310431i\)
\(L(\frac12)\) \(\approx\) \(0.789528 + 0.310431i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.73 - 0.0239i)T \)
7 \( 1 \)
good2 \( 1 + (1.80 - 1.04i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + 3.30T + 5T^{2} \)
11 \( 1 + 2.66iT - 11T^{2} \)
13 \( 1 + (-2.11 + 1.21i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.59 - 6.21i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.24 - 2.45i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 4.99iT - 23T^{2} \)
29 \( 1 + (-5.50 - 3.17i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.30 + 1.33i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.844 + 1.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.553 - 0.958i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.93 + 5.08i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.44 - 4.22i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-8.94 + 5.16i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.56 - 4.44i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.44 - 2.56i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.16 - 7.21i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.07iT - 71T^{2} \)
73 \( 1 + (-6.94 + 4.00i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.50 + 4.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.04 + 1.80i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.541 - 0.937i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (9.47 + 5.46i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.76589888944210946895672390726, −10.21308970883074831117171116862, −8.946585337779717361339433608075, −8.349880177641737842212541733736, −7.910557617444293657590851286322, −7.13814575324891188242492550256, −5.94203283844761010250788204506, −4.04460797765619578953210720661, −3.30582062046366174652572214334, −1.05327153207084667911859901494, 1.06846362592750302253302875612, 2.70086300848972823495140027634, 3.62627688276213140901796826199, 4.88040512669643028989301202993, 7.21889452489793847229346785781, 7.56642516243776230338037713498, 8.388908798557184053211444158829, 9.320610726732840933203260209512, 9.803368217451297201042312339487, 10.95944954433870688182366757149

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