Properties

Label 2-21e2-9.4-c1-0-11
Degree $2$
Conductor $441$
Sign $0.695 + 0.718i$
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  + (−1.08 − 1.88i)2-s + (1.68 + 0.388i)3-s + (−1.36 + 2.36i)4-s + (−0.634 + 1.09i)5-s + (−1.10 − 3.60i)6-s + 1.60·8-s + (2.69 + 1.31i)9-s + 2.76·10-s + (2.73 + 4.74i)11-s + (−3.23 + 3.46i)12-s + (2.37 − 4.10i)13-s + (−1.49 + 1.60i)15-s + (0.992 + 1.71i)16-s − 4.81·17-s + (−0.463 − 6.51i)18-s + 5.38·19-s + ⋯
L(s)  = 1  + (−0.769 − 1.33i)2-s + (0.974 + 0.224i)3-s + (−0.684 + 1.18i)4-s + (−0.283 + 0.491i)5-s + (−0.450 − 1.47i)6-s + 0.566·8-s + (0.899 + 0.437i)9-s + 0.872·10-s + (0.825 + 1.43i)11-s + (−0.932 + 1.00i)12-s + (0.658 − 1.13i)13-s + (−0.386 + 0.415i)15-s + (0.248 + 0.429i)16-s − 1.16·17-s + (−0.109 − 1.53i)18-s + 1.23·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.695 + 0.718i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.695 + 0.718i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18260 - 0.501097i\)
\(L(\frac12)\) \(\approx\) \(1.18260 - 0.501097i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.68 - 0.388i)T \)
7 \( 1 \)
good2 \( 1 + (1.08 + 1.88i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (0.634 - 1.09i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.73 - 4.74i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.37 + 4.10i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.81T + 17T^{2} \)
19 \( 1 - 5.38T + 19T^{2} \)
23 \( 1 + (-2.58 + 4.48i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.01 - 3.49i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.732 + 1.26i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.91T + 37T^{2} \)
41 \( 1 + (1.94 - 3.37i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.66 + 2.87i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.57 + 2.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 7.14T + 53T^{2} \)
59 \( 1 + (0.154 - 0.267i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.17 + 8.95i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.23 - 3.87i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.96T + 71T^{2} \)
73 \( 1 + 10.5T + 73T^{2} \)
79 \( 1 + (-4.50 - 7.80i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.08 + 8.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 5.19T + 89T^{2} \)
97 \( 1 + (2.48 + 4.30i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.79622381786692982285015295326, −10.12926038269411312484803471216, −9.331542748772976153894423214946, −8.672024262707258514932086862558, −7.66604687819454814206512696972, −6.70248824568206736046591804823, −4.69102645269590234892293248626, −3.53960808643213291753063688137, −2.74465740423828460634961363500, −1.47687046560232978014272727038, 1.16112097932892923654049765950, 3.26053224816898564224477424306, 4.51219232036641048179522868664, 6.04617099656342813794849744488, 6.76100144495595726396300693396, 7.68994453273283311977160040717, 8.670220915353595525284759257202, 8.896798363548701652372105194188, 9.681217622856890562596678550153, 11.22592197976994729952055746142

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