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Properties

Label 2-21e2-49.25-c1-0-14
Degree $2$
Conductor $441$
Sign $0.673 + 0.739i$
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  + (−1.16 + 0.358i)2-s + (−0.429 + 0.292i)4-s + (2.56 + 0.386i)5-s + (−0.339 − 2.62i)7-s + (1.91 − 2.39i)8-s + (−3.12 + 0.470i)10-s + (−2.81 − 2.61i)11-s + (−0.203 + 0.891i)13-s + (1.33 + 2.92i)14-s + (−0.982 + 2.50i)16-s + (0.368 − 4.91i)17-s + (−1.24 − 2.15i)19-s + (−1.21 + 0.585i)20-s + (4.21 + 2.02i)22-s + (0.295 + 3.93i)23-s + ⋯
L(s)  = 1  + (−0.822 + 0.253i)2-s + (−0.214 + 0.146i)4-s + (1.14 + 0.172i)5-s + (−0.128 − 0.991i)7-s + (0.675 − 0.847i)8-s + (−0.987 + 0.148i)10-s + (−0.849 − 0.788i)11-s + (−0.0564 + 0.247i)13-s + (0.356 + 0.782i)14-s + (−0.245 + 0.626i)16-s + (0.0894 − 1.19i)17-s + (−0.285 − 0.494i)19-s + (−0.271 + 0.130i)20-s + (0.898 + 0.432i)22-s + (0.0615 + 0.821i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.673 + 0.739i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.673 + 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.673 + 0.739i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.673 + 0.739i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.783565 - 0.346403i\)
\(L(\frac12)\) \(\approx\) \(0.783565 - 0.346403i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.339 + 2.62i)T \)
good2 \( 1 + (1.16 - 0.358i)T + (1.65 - 1.12i)T^{2} \)
5 \( 1 + (-2.56 - 0.386i)T + (4.77 + 1.47i)T^{2} \)
11 \( 1 + (2.81 + 2.61i)T + (0.822 + 10.9i)T^{2} \)
13 \( 1 + (0.203 - 0.891i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 + (-0.368 + 4.91i)T + (-16.8 - 2.53i)T^{2} \)
19 \( 1 + (1.24 + 2.15i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.295 - 3.93i)T + (-22.7 + 3.42i)T^{2} \)
29 \( 1 + (-4.93 + 2.37i)T + (18.0 - 22.6i)T^{2} \)
31 \( 1 + (-4.93 + 8.54i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.81 - 1.23i)T + (13.5 + 34.4i)T^{2} \)
41 \( 1 + (-1.31 + 1.65i)T + (-9.12 - 39.9i)T^{2} \)
43 \( 1 + (2.67 + 3.35i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (-5.46 + 1.68i)T + (38.8 - 26.4i)T^{2} \)
53 \( 1 + (-6.34 + 4.32i)T + (19.3 - 49.3i)T^{2} \)
59 \( 1 + (9.59 - 1.44i)T + (56.3 - 17.3i)T^{2} \)
61 \( 1 + (-11.9 - 8.14i)T + (22.2 + 56.7i)T^{2} \)
67 \( 1 + (3.38 - 5.85i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.32 + 2.56i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (-10.8 - 3.35i)T + (60.3 + 41.1i)T^{2} \)
79 \( 1 + (-2.95 - 5.12i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.604 - 2.64i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (-0.713 + 0.661i)T + (6.65 - 88.7i)T^{2} \)
97 \( 1 + 17.8T + 97T^{2} \)
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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

−10.60788889612746626686064706740, −9.937210333394267702182971792565, −9.363883958869264111878310302680, −8.285335045349918355264373303170, −7.42314911854291236615934691690, −6.55608482693754228181960240109, −5.37017215798164256743598181839, −4.13224487378694495356421317682, −2.63993840035090813661495027671, −0.74616840862363563494493182016, 1.62940835751439127011878280755, 2.61237230132178834141601793421, 4.71665490801536976240170808040, 5.53502736163493732276636799416, 6.45004843927724428443633599860, 8.027499461689862817527202931657, 8.648776105903318885101127956460, 9.535469145706906835479540093504, 10.23985267018542582933575065116, 10.72234535410270577467205236815

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