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Properties

Label 1-21e2-441.110-r0-0-0
Degree $1$
Conductor $441$
Sign $0.944 - 0.328i$
Analytic cond. $2.04799$
Root an. cond. $2.04799$
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  + (0.988 − 0.149i)2-s + (0.955 − 0.294i)4-s + (−0.900 + 0.433i)5-s + (0.900 − 0.433i)8-s + (−0.826 + 0.563i)10-s + (−0.623 − 0.781i)11-s + (0.988 − 0.149i)13-s + (0.826 − 0.563i)16-s + (0.955 + 0.294i)17-s + (0.5 + 0.866i)19-s + (−0.733 + 0.680i)20-s + (−0.733 − 0.680i)22-s + (0.222 − 0.974i)23-s + (0.623 − 0.781i)25-s + (0.955 − 0.294i)26-s + ⋯
L(s)  = 1  + (0.988 − 0.149i)2-s + (0.955 − 0.294i)4-s + (−0.900 + 0.433i)5-s + (0.900 − 0.433i)8-s + (−0.826 + 0.563i)10-s + (−0.623 − 0.781i)11-s + (0.988 − 0.149i)13-s + (0.826 − 0.563i)16-s + (0.955 + 0.294i)17-s + (0.5 + 0.866i)19-s + (−0.733 + 0.680i)20-s + (−0.733 − 0.680i)22-s + (0.222 − 0.974i)23-s + (0.623 − 0.781i)25-s + (0.955 − 0.294i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.944 - 0.328i$
Analytic conductor: \(2.04799\)
Root analytic conductor: \(2.04799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (110, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (0:\ ),\ 0.944 - 0.328i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.262635699 - 0.3823645978i\)
\(L(\frac12)\) \(\approx\) \(2.262635699 - 0.3823645978i\)
\(L(1)\) \(\approx\) \(1.756795588 - 0.1759507745i\)
\(L(1)\) \(\approx\) \(1.756795588 - 0.1759507745i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.988 - 0.149i)T \)
5 \( 1 + (-0.900 + 0.433i)T \)
11 \( 1 + (-0.623 - 0.781i)T \)
13 \( 1 + (0.988 - 0.149i)T \)
17 \( 1 + (0.955 + 0.294i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.222 - 0.974i)T \)
29 \( 1 + (0.733 - 0.680i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (-0.733 + 0.680i)T \)
41 \( 1 + (0.0747 - 0.997i)T \)
43 \( 1 + (0.0747 + 0.997i)T \)
47 \( 1 + (-0.988 + 0.149i)T \)
53 \( 1 + (0.733 + 0.680i)T \)
59 \( 1 + (0.0747 + 0.997i)T \)
61 \( 1 + (-0.955 - 0.294i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.222 - 0.974i)T \)
73 \( 1 + (-0.365 - 0.930i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.988 - 0.149i)T \)
89 \( 1 + (-0.988 - 0.149i)T \)
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

−23.80893516646139473110209348792, −23.29431578072601095956191157859, −22.70594879387624971557101151357, −21.46791758679654584102891447373, −20.767500141415001818137997101521, −20.06507686343503978800316188252, −19.20181030745650105986579573507, −18.02991044858508037645118232788, −16.88773773958883066974912779856, −15.870121799750203377516333742615, −15.57935975950191033844151135299, −14.52343646980272350276317543742, −13.463073598142106089568594452385, −12.764306437283444938566019935103, −11.81455591313837038812182723157, −11.21390695284359374970992661586, −10.0070333894355593354901452414, −8.60484763186056841283688342928, −7.62957474628346761179848744059, −6.9429202339520937020812681103, −5.55638877723957010850035041308, −4.8022560058328119916363138240, −3.791837086192760785940157071060, −2.88667103928103259646759365258, −1.348452535056967044914511827273, 1.18832358395202863558477537732, 2.906129524858665102451192110196, 3.46730062966293712046846872257, 4.54418270163407543031672527784, 5.719542254817866731961433351526, 6.53170675378107750512742918784, 7.728352199192891644659950300644, 8.41393642384188500579291024676, 10.28157764315235791614279827086, 10.783293676698212433227461740840, 11.82191916360826925834922385143, 12.453346970358285641885492517954, 13.61512598461922315809791189702, 14.29191519416521468918438832637, 15.27475644474982768976247072259, 16.00238592570310838638332255066, 16.633774111980960374176870420703, 18.31669356647707114104461566522, 18.97624627918390162500474747663, 19.78884034508949704610725731912, 20.9136904369319711897079828110, 21.27307518877441183939242336589, 22.69675946036974746785193989383, 22.941695231132355824269335618148, 23.837663431911150476198657990323

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