Properties

Degree $2$
Conductor $441$
Sign $0.827 + 0.561i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.20 + 2.09i)2-s + (−1.91 − 3.31i)4-s + (−0.292 + 0.507i)5-s + 4.41·8-s + (−0.707 − 1.22i)10-s + (−1 − 1.73i)11-s − 5.41·13-s + (−1.49 + 2.59i)16-s + (−3.12 − 5.40i)17-s + (1.41 − 2.44i)19-s + 2.24·20-s + 4.82·22-s + (1.82 − 3.16i)23-s + (2.32 + 4.03i)25-s + (6.53 − 11.3i)26-s + ⋯
L(s)  = 1  + (−0.853 + 1.47i)2-s + (−0.957 − 1.65i)4-s + (−0.130 + 0.226i)5-s + 1.56·8-s + (−0.223 − 0.387i)10-s + (−0.301 − 0.522i)11-s − 1.50·13-s + (−0.374 + 0.649i)16-s + (−0.757 − 1.31i)17-s + (0.324 − 0.561i)19-s + 0.501·20-s + 1.02·22-s + (0.381 − 0.660i)23-s + (0.465 + 0.806i)25-s + (1.28 − 2.22i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.827 + 0.561i$
Motivic weight: \(1\)
Character: $\chi_{441} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.827 + 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.335371 - 0.103049i\)
\(L(\frac12)\) \(\approx\) \(0.335371 - 0.103049i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.20 - 2.09i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (0.292 - 0.507i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.41T + 13T^{2} \)
17 \( 1 + (3.12 + 5.40i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.41 + 2.44i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.82 + 3.16i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.17T + 29T^{2} \)
31 \( 1 + (3.41 + 5.91i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.24T + 41T^{2} \)
43 \( 1 + 5.65T + 43T^{2} \)
47 \( 1 + (1.41 - 2.44i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1 + 1.73i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.41 - 5.91i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.87 + 3.25i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.82 + 4.89i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.3T + 71T^{2} \)
73 \( 1 + (2.94 + 5.10i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.17 - 2.02i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 15.3T + 83T^{2} \)
89 \( 1 + (-2.87 + 4.98i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.41T + 97T^{2} \)
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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

−10.80727750820892053434204810990, −9.661443799096982677575971547619, −9.180747058936178411320339737217, −8.135973061998228593406531017041, −7.24270604105199894833649870860, −6.78546102473409594983307933964, −5.47734263462445795433573042490, −4.74158837045971618014781920286, −2.74363793057589466798992633355, −0.28796859418613877262425101429, 1.63312452912655866259965414681, 2.75319720552970535818630212536, 4.01591178117714525488973306410, 5.12590229259933459034359106841, 6.83791026211567828557377316255, 7.976962911792102249882217834644, 8.691570187098303785888912431052, 9.732772895365562233234470000352, 10.22338766037497619362800852580, 11.08442227698367814256430054822

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