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Properties

Label 2-21e2-63.4-c1-0-11
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: $\chi_{441} (67, \cdot )$
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

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

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