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Properties

Label 2-21e2-49.37-c1-0-0
Degree $2$
Conductor $441$
Sign $-0.633 - 0.774i$
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  + (0.968 − 0.659i)2-s + (−0.229 + 0.583i)4-s + (−3.03 − 0.936i)5-s + (−2.50 − 0.844i)7-s + (0.684 + 3.00i)8-s + (−3.55 + 1.09i)10-s + (0.358 + 4.78i)11-s + (−3.44 − 1.65i)13-s + (−2.98 + 0.836i)14-s + (1.72 + 1.59i)16-s + (−0.602 − 0.0908i)17-s + (−1.98 + 3.43i)19-s + (1.24 − 1.55i)20-s + (3.50 + 4.39i)22-s + (−3.22 + 0.486i)23-s + ⋯
L(s)  = 1  + (0.684 − 0.466i)2-s + (−0.114 + 0.291i)4-s + (−1.35 − 0.418i)5-s + (−0.947 − 0.319i)7-s + (0.242 + 1.06i)8-s + (−1.12 + 0.346i)10-s + (0.108 + 1.44i)11-s + (−0.955 − 0.460i)13-s + (−0.797 + 0.223i)14-s + (0.430 + 0.399i)16-s + (−0.146 − 0.0220i)17-s + (−0.454 + 0.787i)19-s + (0.277 − 0.348i)20-s + (0.748 + 0.938i)22-s + (−0.673 + 0.101i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.633 - 0.774i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.633 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.633 - 0.774i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.633 - 0.774i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.183965 + 0.388171i\)
\(L(\frac12)\) \(\approx\) \(0.183965 + 0.388171i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.50 + 0.844i)T \)
good2 \( 1 + (-0.968 + 0.659i)T + (0.730 - 1.86i)T^{2} \)
5 \( 1 + (3.03 + 0.936i)T + (4.13 + 2.81i)T^{2} \)
11 \( 1 + (-0.358 - 4.78i)T + (-10.8 + 1.63i)T^{2} \)
13 \( 1 + (3.44 + 1.65i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 + (0.602 + 0.0908i)T + (16.2 + 5.01i)T^{2} \)
19 \( 1 + (1.98 - 3.43i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.22 - 0.486i)T + (21.9 - 6.77i)T^{2} \)
29 \( 1 + (-0.769 + 0.965i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 + (-0.607 - 1.05i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.77 + 7.07i)T + (-27.1 + 25.1i)T^{2} \)
41 \( 1 + (1.52 + 6.66i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (-0.100 + 0.439i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (-5.92 + 4.04i)T + (17.1 - 43.7i)T^{2} \)
53 \( 1 + (0.873 - 2.22i)T + (-38.8 - 36.0i)T^{2} \)
59 \( 1 + (3.38 - 1.04i)T + (48.7 - 33.2i)T^{2} \)
61 \( 1 + (-4.53 - 11.5i)T + (-44.7 + 41.4i)T^{2} \)
67 \( 1 + (-1.61 - 2.78i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.01 - 6.29i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-2.14 - 1.45i)T + (26.6 + 67.9i)T^{2} \)
79 \( 1 + (2.76 - 4.79i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.63 - 1.27i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (0.623 - 8.32i)T + (-88.0 - 13.2i)T^{2} \)
97 \( 1 - 6.49T + 97T^{2} \)
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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

−11.92127271359908675830543923644, −10.66560797131766583861965012673, −9.799119285217530623863680657078, −8.633786192889379047549790726070, −7.65983794268965527985975716822, −7.07612999971116256001097875214, −5.39482752204982692192195497221, −4.22626224804597261580419649562, −3.83841899685416451535284898156, −2.43184497832745142439148290569, 0.20767090406210835520728034545, 3.00721643421451849344811609348, 3.91581229329752902485946490183, 4.94653175302736623348876361024, 6.23781687917445423317326098972, 6.77245877625785993532195727266, 7.87685817092936297522567458516, 8.943360361182359131053732125729, 9.914875089685768384206401348031, 10.94515742779555513083899520863

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