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Properties

Label 1-21e2-441.169-r0-0-0
Degree $1$
Conductor $441$
Sign $0.999 + 0.0142i$
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 + (0.826 − 0.563i)5-s + (−0.900 − 0.433i)8-s + (−0.900 + 0.433i)10-s + (−0.988 − 0.149i)11-s + (0.365 + 0.930i)13-s + (0.826 + 0.563i)16-s + (−0.222 + 0.974i)17-s + 19-s + (0.955 − 0.294i)20-s + (0.955 + 0.294i)22-s + (0.955 + 0.294i)23-s + (0.365 − 0.930i)25-s + (−0.222 − 0.974i)26-s + ⋯
L(s)  = 1  + (−0.988 − 0.149i)2-s + (0.955 + 0.294i)4-s + (0.826 − 0.563i)5-s + (−0.900 − 0.433i)8-s + (−0.900 + 0.433i)10-s + (−0.988 − 0.149i)11-s + (0.365 + 0.930i)13-s + (0.826 + 0.563i)16-s + (−0.222 + 0.974i)17-s + 19-s + (0.955 − 0.294i)20-s + (0.955 + 0.294i)22-s + (0.955 + 0.294i)23-s + (0.365 − 0.930i)25-s + (−0.222 − 0.974i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0142i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0142i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.999 + 0.0142i$
Analytic conductor: \(2.04799\)
Root analytic conductor: \(2.04799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (0:\ ),\ 0.999 + 0.0142i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9859171947 + 0.007023588690i\)
\(L(\frac12)\) \(\approx\) \(0.9859171947 + 0.007023588690i\)
\(L(1)\) \(\approx\) \(0.8266882059 - 0.04659609260i\)
\(L(1)\) \(\approx\) \(0.8266882059 - 0.04659609260i\)

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 \( 1 + (0.826 - 0.563i)T \)
11 \( 1 + (-0.988 - 0.149i)T \)
13 \( 1 + (0.365 + 0.930i)T \)
17 \( 1 + (-0.222 + 0.974i)T \)
19 \( 1 + T \)
23 \( 1 + (0.955 + 0.294i)T \)
29 \( 1 + (0.955 - 0.294i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.222 + 0.974i)T \)
41 \( 1 + (0.826 - 0.563i)T \)
43 \( 1 + (0.826 + 0.563i)T \)
47 \( 1 + (-0.988 - 0.149i)T \)
53 \( 1 + (-0.222 - 0.974i)T \)
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 \( 1 + (0.623 + 0.781i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.365 - 0.930i)T \)
89 \( 1 + (0.623 + 0.781i)T \)
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

−24.43055201153016284425587489145, −23.186591866895877894233437508235, −22.42361309527917822259208413317, −21.13685844135804370179944747239, −20.67009194586363547907454444496, −19.68482969761476747187331982883, −18.49233808620561141358537536480, −18.12314032859013827508855780894, −17.42948351266725498751494296436, −16.2669707818505595685381928321, −15.565655325390975175592990432066, −14.62648002751844115276007595765, −13.60188863333578914826052170865, −12.601754394665352812089969835360, −11.24594714470755134964183798844, −10.60602868890838324285278083128, −9.77888602494129371519059784504, −8.968968567694585114963715907574, −7.74591368825895974224357846089, −7.07472540751418723248560824627, −5.91152756258942988064574042473, −5.16563545179323961983291708950, −3.05182751588263326284144096924, −2.43287847276986938458649184873, −0.9450893976818680455801359381, 1.13601416493176372745158757341, 2.11073820351450217542220732562, 3.2832816719231090561893654007, 4.89239522424743861177480549526, 5.97869933254140434309050475611, 6.912706102223482359937802000963, 8.11364942230906480660805962401, 8.87110562566288189249586785969, 9.72024418090134104112077387644, 10.55231474335082420382677954827, 11.468395539168986233017713360717, 12.58354472487457261708550785671, 13.36002374155626556754928440048, 14.4818912534910421688514838204, 15.80715137622096308701997105451, 16.297484508285897671611139789135, 17.34991249759097303031735988698, 17.896336438959831349480835183623, 18.84779570673007491168635117140, 19.63855285895793167994437012230, 20.768730759090389173560474581886, 21.15520923635773376298962693783, 21.97989714082397702633737798372, 23.51700988584352297723934312359, 24.23659655718475772958183796349

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