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Properties

Label 1-21e2-441.286-r1-0-0
Degree $1$
Conductor $441$
Sign $-0.267 - 0.963i$
Analytic cond. $47.3920$
Root an. cond. $47.3920$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.955 − 0.294i)2-s + (0.826 − 0.563i)4-s + (−0.365 − 0.930i)5-s + (0.623 − 0.781i)8-s + (−0.623 − 0.781i)10-s + (0.955 − 0.294i)11-s + (0.733 + 0.680i)13-s + (0.365 − 0.930i)16-s + (0.900 − 0.433i)17-s − 19-s + (−0.826 − 0.563i)20-s + (0.826 − 0.563i)22-s + (0.826 − 0.563i)23-s + (−0.733 + 0.680i)25-s + (0.900 + 0.433i)26-s + ⋯
L(s)  = 1  + (0.955 − 0.294i)2-s + (0.826 − 0.563i)4-s + (−0.365 − 0.930i)5-s + (0.623 − 0.781i)8-s + (−0.623 − 0.781i)10-s + (0.955 − 0.294i)11-s + (0.733 + 0.680i)13-s + (0.365 − 0.930i)16-s + (0.900 − 0.433i)17-s − 19-s + (−0.826 − 0.563i)20-s + (0.826 − 0.563i)22-s + (0.826 − 0.563i)23-s + (−0.733 + 0.680i)25-s + (0.900 + 0.433i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.267 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.267 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.267 - 0.963i$
Analytic conductor: \(47.3920\)
Root analytic conductor: \(47.3920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (286, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (1:\ ),\ -0.267 - 0.963i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.398477826 - 3.154741468i\)
\(L(\frac12)\) \(\approx\) \(2.398477826 - 3.154741468i\)
\(L(1)\) \(\approx\) \(1.796886937 - 0.9423496739i\)
\(L(1)\) \(\approx\) \(1.796886937 - 0.9423496739i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.955 - 0.294i)T \)
5 \( 1 + (-0.365 - 0.930i)T \)
11 \( 1 + (0.955 - 0.294i)T \)
13 \( 1 + (0.733 + 0.680i)T \)
17 \( 1 + (0.900 - 0.433i)T \)
19 \( 1 - T \)
23 \( 1 + (0.826 - 0.563i)T \)
29 \( 1 + (0.826 + 0.563i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (-0.900 + 0.433i)T \)
41 \( 1 + (-0.365 - 0.930i)T \)
43 \( 1 + (0.365 - 0.930i)T \)
47 \( 1 + (-0.955 + 0.294i)T \)
53 \( 1 + (-0.900 - 0.433i)T \)
59 \( 1 + (0.988 - 0.149i)T \)
61 \( 1 + (-0.826 - 0.563i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.900 - 0.433i)T \)
73 \( 1 + (0.222 + 0.974i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.733 - 0.680i)T \)
89 \( 1 + (0.222 + 0.974i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.82838495252419091889053772203, −22.97676481283729459208991708039, −22.76998283749066930483988487776, −21.55171325260266116489949938454, −21.0413626806527181148471453376, −19.72197418004895798949445819392, −19.23708353856736268928130236319, −17.87803456291984131187996403474, −17.09442422120158612934465016292, −16.01457518831429927281234604101, −15.14258807571061721118769421250, −14.60521233199587784879574745098, −13.72827907901405002814738059573, −12.68805929650630556088260355675, −11.836922815235193144169638724681, −10.98209080560065870845350507466, −10.1524981165184781963775243786, −8.54261494464698936997862717619, −7.6350141134614833547481994014, −6.620944783563314465789480673446, −6.02823226412305610656276261515, −4.67822511151140119878317708219, −3.632360274683903611592896004341, −2.95472493402238137828967171993, −1.48527209509716336755130642068, 0.79103140124511825908593601504, 1.76289251966985515269254546430, 3.29881424808386004850206793160, 4.17532864153134888154010133235, 4.99777873648258924785993830085, 6.12180686487171212005123143327, 6.988456697851950526196563252, 8.37164437935067125936193428856, 9.23342165166977805004767258758, 10.47217817397501480020981848593, 11.53413715445531328684792988194, 12.11311173721997663365070867891, 12.998607738690303271230791369899, 13.89117614744462915028485542822, 14.68478046886613396136902617010, 15.74858786916993778214093755489, 16.48408683813276459094465675664, 17.20117095718420050738122505726, 18.99379287757635455409737602100, 19.256198826013164257102956456301, 20.55639146799802353827094474520, 20.88501195851468810592593853208, 21.84628090639121172219483543723, 22.84426447549075698152055660512, 23.5493495251280920262429198269

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