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Properties

Label 2-21e2-49.8-c1-0-8
Degree $2$
Conductor $441$
Sign $0.227 + 0.973i$
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  + (−1.34 − 1.68i)2-s + (−0.586 + 2.57i)4-s + (2.99 − 1.44i)5-s + (2.53 + 0.746i)7-s + (1.23 − 0.594i)8-s + (−6.44 − 3.10i)10-s + (0.574 + 0.720i)11-s + (1.96 + 2.46i)13-s + (−2.15 − 5.27i)14-s + (2.09 + 1.00i)16-s + (1.19 + 5.22i)17-s + 5.32·19-s + (1.94 + 8.53i)20-s + (0.441 − 1.93i)22-s + (−0.320 + 1.40i)23-s + ⋯
L(s)  = 1  + (−0.949 − 1.19i)2-s + (−0.293 + 1.28i)4-s + (1.33 − 0.644i)5-s + (0.959 + 0.282i)7-s + (0.436 − 0.210i)8-s + (−2.03 − 0.981i)10-s + (0.173 + 0.217i)11-s + (0.545 + 0.684i)13-s + (−0.574 − 1.40i)14-s + (0.523 + 0.251i)16-s + (0.289 + 1.26i)17-s + 1.22·19-s + (0.435 + 1.90i)20-s + (0.0940 − 0.412i)22-s + (−0.0668 + 0.292i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.227 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.227 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.227 + 0.973i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.227 + 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.936037 - 0.742653i\)
\(L(\frac12)\) \(\approx\) \(0.936037 - 0.742653i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-2.53 - 0.746i)T \)
good2 \( 1 + (1.34 + 1.68i)T + (-0.445 + 1.94i)T^{2} \)
5 \( 1 + (-2.99 + 1.44i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (-0.574 - 0.720i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (-1.96 - 2.46i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 + (-1.19 - 5.22i)T + (-15.3 + 7.37i)T^{2} \)
19 \( 1 - 5.32T + 19T^{2} \)
23 \( 1 + (0.320 - 1.40i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (0.472 + 2.06i)T + (-26.1 + 12.5i)T^{2} \)
31 \( 1 + 9.98T + 31T^{2} \)
37 \( 1 + (1.87 + 8.19i)T + (-33.3 + 16.0i)T^{2} \)
41 \( 1 + (6.36 - 3.06i)T + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (3.89 + 1.87i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (-2.59 - 3.25i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (-2.21 + 9.72i)T + (-47.7 - 22.9i)T^{2} \)
59 \( 1 + (6.17 + 2.97i)T + (36.7 + 46.1i)T^{2} \)
61 \( 1 + (1.19 + 5.21i)T + (-54.9 + 26.4i)T^{2} \)
67 \( 1 + 8.94T + 67T^{2} \)
71 \( 1 + (1.42 - 6.22i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-8.70 + 10.9i)T + (-16.2 - 71.1i)T^{2} \)
79 \( 1 + 13.1T + 79T^{2} \)
83 \( 1 + (-9.25 + 11.6i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (0.756 - 0.948i)T + (-19.8 - 86.7i)T^{2} \)
97 \( 1 - 1.28T + 97T^{2} \)
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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

−10.83931250645893443187962695751, −9.987391122927754890049084503988, −9.202997254072004786654567425375, −8.725422182947617674209549218551, −7.66538401092274718676841427770, −6.02528449320794012931578149546, −5.19810313873064545962470975701, −3.66281066923177554213742742285, −1.93136532605402958644574263688, −1.52171751283282706818958690247, 1.33868550469522420156426206126, 3.10130673069830470492582495873, 5.22138326600154357009173319322, 5.76292533118829791670626682349, 6.90584720241506544179767538437, 7.50979333885004615894800273903, 8.563221631370493342683839196856, 9.383300455160135128400473026758, 10.14938220239848981547961945011, 10.90921957266167839163055592281

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