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Properties

Label 2-21e2-49.8-c1-0-18
Degree $2$
Conductor $441$
Sign $0.750 + 0.660i$
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  + (0.385 + 0.483i)2-s + (0.359 − 1.57i)4-s + (3.63 − 1.75i)5-s + (−2.64 + 0.0631i)7-s + (2.01 − 0.970i)8-s + (2.25 + 1.08i)10-s + (−0.0332 − 0.0416i)11-s + (−0.237 − 0.297i)13-s + (−1.05 − 1.25i)14-s + (−1.66 − 0.803i)16-s + (0.172 + 0.755i)17-s − 4.32·19-s + (−1.45 − 6.36i)20-s + (0.00733 − 0.0321i)22-s + (0.445 − 1.95i)23-s + ⋯
L(s)  = 1  + (0.272 + 0.341i)2-s + (0.179 − 0.788i)4-s + (1.62 − 0.783i)5-s + (−0.999 + 0.0238i)7-s + (0.712 − 0.343i)8-s + (0.711 + 0.342i)10-s + (−0.0100 − 0.0125i)11-s + (−0.0657 − 0.0825i)13-s + (−0.280 − 0.335i)14-s + (−0.416 − 0.200i)16-s + (0.0418 + 0.183i)17-s − 0.991·19-s + (−0.325 − 1.42i)20-s + (0.00156 − 0.00685i)22-s + (0.0928 − 0.406i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.750 + 0.660i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.750 + 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.750 + 0.660i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.750 + 0.660i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83743 - 0.693317i\)
\(L(\frac12)\) \(\approx\) \(1.83743 - 0.693317i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.64 - 0.0631i)T \)
good2 \( 1 + (-0.385 - 0.483i)T + (-0.445 + 1.94i)T^{2} \)
5 \( 1 + (-3.63 + 1.75i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (0.0332 + 0.0416i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (0.237 + 0.297i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 + (-0.172 - 0.755i)T + (-15.3 + 7.37i)T^{2} \)
19 \( 1 + 4.32T + 19T^{2} \)
23 \( 1 + (-0.445 + 1.95i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (-1.94 - 8.52i)T + (-26.1 + 12.5i)T^{2} \)
31 \( 1 - 5.67T + 31T^{2} \)
37 \( 1 + (-2.39 - 10.5i)T + (-33.3 + 16.0i)T^{2} \)
41 \( 1 + (-1.87 + 0.902i)T + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (6.37 + 3.06i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (-6.67 - 8.37i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (-1.51 + 6.65i)T + (-47.7 - 22.9i)T^{2} \)
59 \( 1 + (-6.38 - 3.07i)T + (36.7 + 46.1i)T^{2} \)
61 \( 1 + (-1.24 - 5.45i)T + (-54.9 + 26.4i)T^{2} \)
67 \( 1 + 10.4T + 67T^{2} \)
71 \( 1 + (-2.39 + 10.5i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (1.54 - 1.93i)T + (-16.2 - 71.1i)T^{2} \)
79 \( 1 + 10.7T + 79T^{2} \)
83 \( 1 + (0.559 - 0.702i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (7.55 - 9.47i)T + (-19.8 - 86.7i)T^{2} \)
97 \( 1 - 9.99T + 97T^{2} \)
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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

−10.55375167025449443632181013470, −10.15476238639135458267700341032, −9.344037702893879038369267074000, −8.543916862275409285831622867792, −6.78028382938088999134999031380, −6.27171776400062195066405952158, −5.44700284721279833648906536314, −4.54279685941609488191967095812, −2.61754195430856721838693428846, −1.27778988004045780094656518874, 2.19230018957273858942954584705, 2.89295037015346638697505167023, 4.16047122549388479368955524834, 5.72325556406774210198942667476, 6.49976350605793421419848105696, 7.30334910223005281855616969208, 8.657985641798439557193424493070, 9.677882275915819630141453674497, 10.27532383811289320911524986373, 11.16416272078696991571345301088

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