Properties

Label 2-21e2-63.16-c1-0-15
Degree $2$
Conductor $441$
Sign $0.469 + 0.882i$
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  + (−0.649 − 1.12i)2-s + (−1.52 + 0.821i)3-s + (0.155 − 0.268i)4-s + 3.52·5-s + (1.91 + 1.18i)6-s − 3.00·8-s + (1.65 − 2.50i)9-s + (−2.29 − 3.96i)10-s + 1.17·11-s + (−0.0159 + 0.537i)12-s + (1.61 + 2.78i)13-s + (−5.37 + 2.89i)15-s + (1.64 + 2.84i)16-s + (2.45 + 4.24i)17-s + (−3.89 − 0.231i)18-s + (3.43 − 5.94i)19-s + ⋯
L(s)  = 1  + (−0.459 − 0.796i)2-s + (−0.880 + 0.474i)3-s + (0.0775 − 0.134i)4-s + 1.57·5-s + (0.782 + 0.482i)6-s − 1.06·8-s + (0.550 − 0.834i)9-s + (−0.724 − 1.25i)10-s + 0.355·11-s + (−0.00460 + 0.155i)12-s + (0.446 + 0.773i)13-s + (−1.38 + 0.747i)15-s + (0.410 + 0.710i)16-s + (0.594 + 1.02i)17-s + (−0.917 − 0.0545i)18-s + (0.787 − 1.36i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.469 + 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.469 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.469 + 0.882i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.469 + 0.882i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.996504 - 0.598499i\)
\(L(\frac12)\) \(\approx\) \(0.996504 - 0.598499i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.52 - 0.821i)T \)
7 \( 1 \)
good2 \( 1 + (0.649 + 1.12i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 - 3.52T + 5T^{2} \)
11 \( 1 - 1.17T + 11T^{2} \)
13 \( 1 + (-1.61 - 2.78i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.45 - 4.24i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.43 + 5.94i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 4.29T + 23T^{2} \)
29 \( 1 + (-1.36 + 2.35i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.960 + 1.66i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.88 + 8.45i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.32 + 5.76i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.83 + 8.37i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.316 + 0.548i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.11 - 1.92i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.10 - 7.11i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.82 - 8.36i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.66 - 4.61i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.27T + 71T^{2} \)
73 \( 1 + (0.519 + 0.898i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.502 + 0.869i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.65 - 6.33i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (6.02 - 10.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.46 - 9.46i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.75818183972522149546793063896, −10.18582657878746369272146407313, −9.435314419552711712471332002166, −8.921809089762505626555982941882, −6.91404235602459417397158984822, −5.98393744347563198300399857936, −5.54688024858633525346714260065, −3.99825896168361611370539203182, −2.34293012732265238891383011118, −1.16524525279113982301832165775, 1.38511385802921453344496488809, 3.00447230935364768705198705554, 5.11034204687438763521705067113, 5.99263876251340300446758826012, 6.37372453840497227320676817877, 7.52811280790836733450104702505, 8.299101902748930011780956087153, 9.690110444335860734117414678946, 10.01706791652630146043108183444, 11.33695196765075594481804222774

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