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Properties

Label 2-21e2-21.17-c1-0-12
Degree $2$
Conductor $441$
Sign $0.0285 + 0.999i$
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  + (2.23 − 1.28i)2-s + (2.32 − 4.02i)4-s − 6.82i·8-s + (0.790 + 0.456i)11-s + (−4.14 − 7.18i)16-s + 2.35·22-s + (−8.13 + 4.69i)23-s + (2.5 − 4.33i)25-s + 6.06i·29-s + (−6.69 − 3.86i)32-s + (5.29 + 9.16i)37-s + 5.29·43-s + (3.67 − 2.12i)44-s + (−12.1 + 20.9i)46-s − 12.8i·50-s + ⋯
L(s)  = 1  + (1.57 − 0.911i)2-s + (1.16 − 2.01i)4-s − 2.41i·8-s + (0.238 + 0.137i)11-s + (−1.03 − 1.79i)16-s + 0.501·22-s + (−1.69 + 0.979i)23-s + (0.5 − 0.866i)25-s + 1.12i·29-s + (−1.18 − 0.683i)32-s + (0.869 + 1.50i)37-s + 0.806·43-s + (0.553 − 0.319i)44-s + (−1.78 + 3.09i)46-s − 1.82i·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0285 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0285 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.0285 + 0.999i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.0285 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.31225 - 2.24726i\)
\(L(\frac12)\) \(\approx\) \(2.31225 - 2.24726i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-2.23 + 1.28i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.790 - 0.456i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (8.13 - 4.69i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.06iT - 29T^{2} \)
31 \( 1 + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.29 - 9.16i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 5.29T + 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (12.6 + 7.27i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.57iT - 71T^{2} \)
73 \( 1 + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−11.20782057547757116208955585373, −10.30771217226886529720974722337, −9.544717054436425274719918877107, −8.085336250260808361552746896951, −6.71523008614981429990611822990, −5.88935507562271143202426488475, −4.86215947201267407264216808373, −3.96280895594238625105097875586, −2.90773189180989679630659992956, −1.61791347771318754566543672047, 2.48022352496112040776232340583, 3.79309202577238186228373892707, 4.56223652706810401582641653739, 5.75353887780185290949511411492, 6.34580069504739126241985885711, 7.43647426211908797899810216914, 8.159219424732148407973000618502, 9.409495223584358151919849856003, 10.77512341722589454502067137243, 11.71048052141046323907089397950

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