Properties

Label 2-21e2-441.142-c1-0-35
Degree $2$
Conductor $441$
Sign $0.221 - 0.975i$
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  + (−0.570 + 1.45i)2-s + (1.65 + 0.515i)3-s + (−0.321 − 0.298i)4-s + (2.55 − 1.22i)5-s + (−1.69 + 2.10i)6-s + (2.20 − 1.46i)7-s + (−2.19 + 1.05i)8-s + (2.46 + 1.70i)9-s + (0.330 + 4.40i)10-s + (−2.88 − 3.61i)11-s + (−0.378 − 0.659i)12-s + (−1.70 + 4.33i)13-s + (0.876 + 4.03i)14-s + (4.85 − 0.715i)15-s + (−0.350 − 4.67i)16-s + (4.28 − 3.97i)17-s + ⋯
L(s)  = 1  + (−0.403 + 1.02i)2-s + (0.954 + 0.297i)3-s + (−0.160 − 0.149i)4-s + (1.14 − 0.549i)5-s + (−0.691 + 0.861i)6-s + (0.832 − 0.554i)7-s + (−0.776 + 0.373i)8-s + (0.822 + 0.568i)9-s + (0.104 + 1.39i)10-s + (−0.869 − 1.08i)11-s + (−0.109 − 0.190i)12-s + (−0.471 + 1.20i)13-s + (0.234 + 1.07i)14-s + (1.25 − 0.184i)15-s + (−0.0875 − 1.16i)16-s + (1.04 − 0.965i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.221 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.221 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.221 - 0.975i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.221 - 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52633 + 1.21872i\)
\(L(\frac12)\) \(\approx\) \(1.52633 + 1.21872i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.65 - 0.515i)T \)
7 \( 1 + (-2.20 + 1.46i)T \)
good2 \( 1 + (0.570 - 1.45i)T + (-1.46 - 1.36i)T^{2} \)
5 \( 1 + (-2.55 + 1.22i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (2.88 + 3.61i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (1.70 - 4.33i)T + (-9.52 - 8.84i)T^{2} \)
17 \( 1 + (-4.28 + 3.97i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (0.412 - 0.714i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.126 - 0.552i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (2.40 + 0.741i)T + (23.9 + 16.3i)T^{2} \)
31 \( 1 + (3.94 - 6.83i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.13 + 0.349i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (10.1 + 6.90i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (0.809 - 0.551i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (2.96 - 7.56i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (5.50 - 1.69i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (5.21 - 3.55i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (-3.86 + 3.58i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (3.39 - 5.87i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.13 + 4.96i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-2.74 - 0.413i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (-3.90 - 6.75i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.49 + 8.89i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (-4.37 - 11.1i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (-2.19 + 3.79i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.11039619594470606761835904315, −10.02262464026902966757121937205, −9.225315324468298366899257308596, −8.554994908105244633601776884174, −7.73209754150137851398250808748, −6.96888744689688428848270911555, −5.58142427276032796536524197715, −4.89407540797101943139797370078, −3.19706218363182718577220988489, −1.81137885114042506951169451891, 1.75179447290671527233856786024, 2.31461273842977329927123571081, 3.32411922107453933220801443812, 5.16650686470487224413967057878, 6.22002085337674781139564027257, 7.54232031817328408990334701011, 8.316086763587529895417480408532, 9.485014268668854135805017649642, 10.09719891374499684961652007660, 10.53636488671439408868915994293

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