Properties

Label 2-21e2-49.15-c1-0-22
Degree $2$
Conductor $441$
Sign $-0.831 - 0.555i$
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  + (0.569 − 2.49i)2-s + (−4.09 − 1.97i)4-s + (1.87 − 2.34i)5-s + (−2.56 + 0.655i)7-s + (−4.05 + 5.08i)8-s + (−4.79 − 6.01i)10-s + (1.00 − 4.38i)11-s + (−0.396 + 1.73i)13-s + (0.175 + 6.76i)14-s + (4.69 + 5.89i)16-s + (−4.12 + 1.98i)17-s + 4.76·19-s + (−12.2 + 5.92i)20-s + (−10.3 − 4.98i)22-s + (−3.59 − 1.73i)23-s + ⋯
L(s)  = 1  + (0.402 − 1.76i)2-s + (−2.04 − 0.985i)4-s + (0.838 − 1.05i)5-s + (−0.968 + 0.247i)7-s + (−1.43 + 1.79i)8-s + (−1.51 − 1.90i)10-s + (0.301 − 1.32i)11-s + (−0.109 + 0.481i)13-s + (0.0469 + 1.80i)14-s + (1.17 + 1.47i)16-s + (−1.00 + 0.482i)17-s + 1.09·19-s + (−2.74 + 1.32i)20-s + (−2.20 − 1.06i)22-s + (−0.750 − 0.361i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.831 - 0.555i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.831 - 0.555i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.831 - 0.555i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.406098 + 1.33843i\)
\(L(\frac12)\) \(\approx\) \(0.406098 + 1.33843i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.56 - 0.655i)T \)
good2 \( 1 + (-0.569 + 2.49i)T + (-1.80 - 0.867i)T^{2} \)
5 \( 1 + (-1.87 + 2.34i)T + (-1.11 - 4.87i)T^{2} \)
11 \( 1 + (-1.00 + 4.38i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (0.396 - 1.73i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 + (4.12 - 1.98i)T + (10.5 - 13.2i)T^{2} \)
19 \( 1 - 4.76T + 19T^{2} \)
23 \( 1 + (3.59 + 1.73i)T + (14.3 + 17.9i)T^{2} \)
29 \( 1 + (-6.40 + 3.08i)T + (18.0 - 22.6i)T^{2} \)
31 \( 1 - 4.69T + 31T^{2} \)
37 \( 1 + (6.00 - 2.89i)T + (23.0 - 28.9i)T^{2} \)
41 \( 1 + (-2.91 + 3.65i)T + (-9.12 - 39.9i)T^{2} \)
43 \( 1 + (0.525 + 0.658i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (-0.612 + 2.68i)T + (-42.3 - 20.3i)T^{2} \)
53 \( 1 + (-11.2 - 5.42i)T + (33.0 + 41.4i)T^{2} \)
59 \( 1 + (3.57 + 4.48i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (-11.6 + 5.60i)T + (38.0 - 47.6i)T^{2} \)
67 \( 1 + 2.84T + 67T^{2} \)
71 \( 1 + (12.5 + 6.05i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (1.80 + 7.92i)T + (-65.7 + 31.6i)T^{2} \)
79 \( 1 - 4.32T + 79T^{2} \)
83 \( 1 + (-0.202 - 0.888i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (-0.265 - 1.16i)T + (-80.1 + 38.6i)T^{2} \)
97 \( 1 - 10.3T + 97T^{2} \)
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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

−10.55390896536053055332284853078, −9.874502277161403845458807465897, −9.041893845977023132201519991259, −8.594740934299516884210396085658, −6.37475360629642500089990747668, −5.51789992244922789108426194446, −4.43715569504904250424274392028, −3.34307745941173391457871945530, −2.18316117687055955143614771228, −0.797458104651039131465067261026, 2.77923442408679383150743712194, 4.14615578432755315952032719946, 5.31327897990973169305888588652, 6.29389691950020868150788869167, 6.92292282049041041261278965944, 7.42985790182562991101132296052, 8.818485778253026675799179892632, 9.783810317547131705271689636231, 10.25777844170181600640604647881, 11.92795723619662459062255888236

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