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Properties

Label 1-21e2-441.400-r0-0-0
Degree $1$
Conductor $441$
Sign $-0.996 - 0.0782i$
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  + (−0.733 − 0.680i)2-s + (0.0747 + 0.997i)4-s + (−0.988 − 0.149i)5-s + (0.623 − 0.781i)8-s + (0.623 + 0.781i)10-s + (−0.733 − 0.680i)11-s + (0.955 − 0.294i)13-s + (−0.988 + 0.149i)16-s + (−0.900 + 0.433i)17-s + 19-s + (0.0747 − 0.997i)20-s + (0.0747 + 0.997i)22-s + (0.0747 + 0.997i)23-s + (0.955 + 0.294i)25-s + (−0.900 − 0.433i)26-s + ⋯
L(s)  = 1  + (−0.733 − 0.680i)2-s + (0.0747 + 0.997i)4-s + (−0.988 − 0.149i)5-s + (0.623 − 0.781i)8-s + (0.623 + 0.781i)10-s + (−0.733 − 0.680i)11-s + (0.955 − 0.294i)13-s + (−0.988 + 0.149i)16-s + (−0.900 + 0.433i)17-s + 19-s + (0.0747 − 0.997i)20-s + (0.0747 + 0.997i)22-s + (0.0747 + 0.997i)23-s + (0.955 + 0.294i)25-s + (−0.900 − 0.433i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0782i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.996 - 0.0782i$
Analytic conductor: \(2.04799\)
Root analytic conductor: \(2.04799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (400, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (0:\ ),\ -0.996 - 0.0782i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01174611201 - 0.2996386636i\)
\(L(\frac12)\) \(\approx\) \(0.01174611201 - 0.2996386636i\)
\(L(1)\) \(\approx\) \(0.4770235551 - 0.2131557858i\)
\(L(1)\) \(\approx\) \(0.4770235551 - 0.2131557858i\)

Euler product

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

−24.463946960964592139081293586510, −23.67650946681000294060845987689, −23.06992329704603884787438227721, −22.22855282879026612193836074328, −20.57458912484146936300711053010, −20.17536821490828819294302510838, −19.12371507179584188358877233403, −18.277161630535907605565850380409, −17.831808403893075179504560682618, −16.32911786301602436794746971318, −15.99791560779984869575391381522, −15.15610332653302793628932931299, −14.27495141451850993543961218802, −13.185467225708299385644059619079, −11.93544996273419091061987037564, −10.98481456854913647689996913184, −10.27720076445223391904942201904, −8.98167255604126339457844711989, −8.347166996095770703586284363952, −7.25509272894231102412035709778, −6.735028700066765433518132071931, −5.30598831820565979144925050205, −4.4195623772089149978015682187, −2.97182269400150856189480455718, −1.44742811439305223573654944182, 0.22860656607232158270735438958, 1.65029645760759714089652934043, 3.16789587719542025846444014104, 3.75629366937607698015188862954, 5.09671535198384743366041740093, 6.58480051192499536743681084224, 7.8188562661677309676282004438, 8.27241784750294370480435369845, 9.26940073815792147416031164479, 10.43060823907056872668093823842, 11.28942246902808257467902683596, 11.77609802361960088364879870096, 13.05936007409861673275963473663, 13.568315139699916537257499794491, 15.37241881939492259872547025804, 15.837575042722157652573065950710, 16.750130880573534786066896351941, 17.81392253945560068798976100587, 18.62036120914470614136955024004, 19.28946034637798397831273628482, 20.204880585356400457587333366465, 20.77663098631416146935489531330, 21.79971441725708869464771293175, 22.67418693104571133498665543687, 23.64855573266822176890879451380

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