Properties

Label 2-21e2-441.142-c1-0-33
Degree $2$
Conductor $441$
Sign $0.0566 + 0.998i$
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  + (−0.342 + 0.873i)2-s + (−1.22 + 1.22i)3-s + (0.820 + 0.760i)4-s + (−2.38 + 1.15i)5-s + (−0.653 − 1.48i)6-s + (0.937 − 2.47i)7-s + (−2.63 + 1.27i)8-s + (−0.0142 − 2.99i)9-s + (−0.186 − 2.48i)10-s + (−3.42 − 4.29i)11-s + (−1.93 + 0.0770i)12-s + (−1.37 + 3.50i)13-s + (1.84 + 1.66i)14-s + (1.50 − 4.33i)15-s + (−0.0381 − 0.509i)16-s + (−2.12 + 1.97i)17-s + ⋯
L(s)  = 1  + (−0.242 + 0.617i)2-s + (−0.705 + 0.708i)3-s + (0.410 + 0.380i)4-s + (−1.06 + 0.514i)5-s + (−0.266 − 0.607i)6-s + (0.354 − 0.935i)7-s + (−0.932 + 0.449i)8-s + (−0.00474 − 0.999i)9-s + (−0.0588 − 0.785i)10-s + (−1.03 − 1.29i)11-s + (−0.558 + 0.0222i)12-s + (−0.381 + 0.970i)13-s + (0.491 + 0.445i)14-s + (0.389 − 1.12i)15-s + (−0.00954 − 0.127i)16-s + (−0.515 + 0.477i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0566 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0566 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.0566 + 0.998i$
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: \((2,\ 441,\ (\ :1/2),\ 0.0566 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0106985 - 0.0101086i\)
\(L(\frac12)\) \(\approx\) \(0.0106985 - 0.0101086i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.22 - 1.22i)T \)
7 \( 1 + (-0.937 + 2.47i)T \)
good2 \( 1 + (0.342 - 0.873i)T + (-1.46 - 1.36i)T^{2} \)
5 \( 1 + (2.38 - 1.15i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (3.42 + 4.29i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (1.37 - 3.50i)T + (-9.52 - 8.84i)T^{2} \)
17 \( 1 + (2.12 - 1.97i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (-1.07 + 1.86i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.11 + 4.90i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (-5.56 - 1.71i)T + (23.9 + 16.3i)T^{2} \)
31 \( 1 + (-0.882 + 1.52i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (9.91 + 3.05i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (3.64 + 2.48i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (8.40 - 5.72i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (-0.573 + 1.46i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (-3.11 + 0.961i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (2.90 - 1.98i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (3.65 - 3.38i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (-3.22 + 5.58i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.06 - 9.04i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (0.706 + 0.106i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (5.41 + 9.37i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.20 - 15.8i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (2.89 + 7.37i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (-0.441 + 0.765i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.96397259702011244381803714872, −10.35369062128251301537844144174, −8.794760565850179006361458454688, −8.110763827475682924218092095711, −7.05125324502337489981736296840, −6.52095682826842900131489554072, −5.14121650623397590226825020831, −4.02405781497066019488309613310, −3.04939626861136364880214491769, −0.01033249007351956058151560673, 1.71260266001591956998157821031, 2.88235283669083358545369515084, 4.89322648809041445350479324606, 5.43289696469044306269260209312, 6.81562236834223492348427175684, 7.69314359352178571630062079023, 8.470288445047992056608612694919, 9.848878677028602182919742345789, 10.57975140670981173769978816591, 11.57051142412283312885621784324

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