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Properties

Label 2-21e2-63.47-c1-0-16
Degree $2$
Conductor $441$
Sign $0.668 + 0.743i$
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  + (0.367 − 0.212i)2-s + (−1.71 − 0.266i)3-s + (−0.910 + 1.57i)4-s − 3.60·5-s + (−0.685 + 0.265i)6-s + 1.62i·8-s + (2.85 + 0.911i)9-s + (−1.32 + 0.765i)10-s − 3.70i·11-s + (1.97 − 2.45i)12-s + (5.23 − 3.02i)13-s + (6.17 + 0.960i)15-s + (−1.47 − 2.55i)16-s + (0.532 + 0.921i)17-s + (1.24 − 0.271i)18-s + (3.16 + 1.82i)19-s + ⋯
L(s)  = 1  + (0.259 − 0.149i)2-s + (−0.988 − 0.153i)3-s + (−0.455 + 0.788i)4-s − 1.61·5-s + (−0.279 + 0.108i)6-s + 0.572i·8-s + (0.952 + 0.303i)9-s + (−0.419 + 0.241i)10-s − 1.11i·11-s + (0.570 − 0.708i)12-s + (1.45 − 0.838i)13-s + (1.59 + 0.248i)15-s + (−0.369 − 0.639i)16-s + (0.129 + 0.223i)17-s + (0.293 − 0.0639i)18-s + (0.725 + 0.418i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.668 + 0.743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.668 + 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.668 + 0.743i$
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: \((2,\ 441,\ (\ :1/2),\ 0.668 + 0.743i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.629090 - 0.280363i\)
\(L(\frac12)\) \(\approx\) \(0.629090 - 0.280363i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.71 + 0.266i)T \)
7 \( 1 \)
good2 \( 1 + (-0.367 + 0.212i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + 3.60T + 5T^{2} \)
11 \( 1 + 3.70iT - 11T^{2} \)
13 \( 1 + (-5.23 + 3.02i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.532 - 0.921i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.16 - 1.82i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 0.363iT - 23T^{2} \)
29 \( 1 + (0.857 + 0.495i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.939 + 0.542i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.00 + 6.93i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.09 + 3.62i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.89 - 3.28i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.83 - 4.91i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.92 + 2.26i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.62 + 9.74i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.0238 - 0.0137i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.86 + 8.42i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.55iT - 71T^{2} \)
73 \( 1 + (-1.95 + 1.12i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.26 + 5.65i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.52 + 2.64i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.47 + 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.67 - 0.964i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.32509331912532950267347790073, −10.57763663698542189131642553785, −8.947553209077153593094716083176, −8.015124374236467929943522862832, −7.59242113872741590931420702750, −6.17793284233098949934076055890, −5.15833560329796851270994023103, −3.89152522717518590481734074778, −3.42554132621871373889781798361, −0.60498501205554544628854919617, 1.10426209298129712351910414997, 3.83745128248536672414497376103, 4.42985343927129874223513720907, 5.36821999439469283568698559463, 6.60973970286583362548826574644, 7.24099795214927888931405847820, 8.548501629801300253486658756613, 9.582972503412802427764693082933, 10.49789570493024427536203127733, 11.42302734123776170305231072524

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