Properties

Label 2-21e2-49.15-c1-0-19
Degree $2$
Conductor $441$
Sign $-0.967 + 0.253i$
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  + (0.5 − 2.19i)2-s + (−2.74 − 1.32i)4-s + (−0.153 + 0.193i)5-s + (2.06 − 1.64i)7-s + (−1.46 + 1.84i)8-s + (0.346 + 0.433i)10-s + (0.233 − 1.02i)11-s + (1.18 − 5.21i)13-s + (−2.57 − 5.35i)14-s + (−0.499 − 0.626i)16-s + (−4.44 + 2.14i)17-s − 5.85·19-s + (0.678 − 0.326i)20-s + (−2.12 − 1.02i)22-s + (5.44 + 2.62i)23-s + ⋯
L(s)  = 1  + (0.353 − 1.54i)2-s + (−1.37 − 0.661i)4-s + (−0.0688 + 0.0863i)5-s + (0.781 − 0.623i)7-s + (−0.519 + 0.651i)8-s + (0.109 + 0.137i)10-s + (0.0703 − 0.308i)11-s + (0.329 − 1.44i)13-s + (−0.689 − 1.43i)14-s + (−0.124 − 0.156i)16-s + (−1.07 + 0.519i)17-s − 1.34·19-s + (0.151 − 0.0730i)20-s + (−0.452 − 0.218i)22-s + (1.13 + 0.547i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 + 0.253i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.967 + 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.967 + 0.253i$
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: \((2,\ 441,\ (\ :1/2),\ -0.967 + 0.253i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.203975 - 1.58199i\)
\(L(\frac12)\) \(\approx\) \(0.203975 - 1.58199i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.06 + 1.64i)T \)
good2 \( 1 + (-0.5 + 2.19i)T + (-1.80 - 0.867i)T^{2} \)
5 \( 1 + (0.153 - 0.193i)T + (-1.11 - 4.87i)T^{2} \)
11 \( 1 + (-0.233 + 1.02i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (-1.18 + 5.21i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 + (4.44 - 2.14i)T + (10.5 - 13.2i)T^{2} \)
19 \( 1 + 5.85T + 19T^{2} \)
23 \( 1 + (-5.44 - 2.62i)T + (14.3 + 17.9i)T^{2} \)
29 \( 1 + (-2.04 + 0.984i)T + (18.0 - 22.6i)T^{2} \)
31 \( 1 + 0.198T + 31T^{2} \)
37 \( 1 + (-5.29 + 2.55i)T + (23.0 - 28.9i)T^{2} \)
41 \( 1 + (-3.04 + 3.82i)T + (-9.12 - 39.9i)T^{2} \)
43 \( 1 + (-2.67 - 3.35i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (-1.82 + 7.98i)T + (-42.3 - 20.3i)T^{2} \)
53 \( 1 + (-3.08 - 1.48i)T + (33.0 + 41.4i)T^{2} \)
59 \( 1 + (-2.96 - 3.71i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (2.96 - 1.43i)T + (38.0 - 47.6i)T^{2} \)
67 \( 1 + 3.35T + 67T^{2} \)
71 \( 1 + (3.65 + 1.76i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (-3.09 - 13.5i)T + (-65.7 + 31.6i)T^{2} \)
79 \( 1 - 8.64T + 79T^{2} \)
83 \( 1 + (-2.62 - 11.4i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (-3.70 - 16.2i)T + (-80.1 + 38.6i)T^{2} \)
97 \( 1 + 1.69T + 97T^{2} \)
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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

−10.93005957864125569213812380990, −10.33412877559778163058203574962, −9.100753029019350421265930390927, −8.220465570918594448069031802242, −7.04112677032438564281992572222, −5.56298069381681777774212712291, −4.46522111264836069283533552594, −3.60376678114495628757972271468, −2.37783180492650320097812825296, −0.969023841601503680259129312344, 2.21765084941969900090748997822, 4.50770393053126374422946010113, 4.65407160808077856731974626160, 6.18292659323377546278393413535, 6.68178683507783532879022748809, 7.74117849830941942465615617110, 8.783424717515976324637810830139, 9.040430127105724998389518363766, 10.77073548170395511236357166546, 11.55907789839447564817886218517

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