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Properties

Label 1-21e2-441.92-r1-0-0
Degree $1$
Conductor $441$
Sign $-0.0782 - 0.996i$
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  + (0.733 − 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

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

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.0782 - 0.996i$
Analytic conductor: \(47.3920\)
Root analytic conductor: \(47.3920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (92, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (1:\ ),\ -0.0782 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.908171778 - 3.145480533i\)
\(L(\frac12)\) \(\approx\) \(2.908171778 - 3.145480533i\)
\(L(1)\) \(\approx\) \(1.808205654 - 1.061941301i\)
\(L(1)\) \(\approx\) \(1.808205654 - 1.061941301i\)

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

−24.22475090329439794549635343879, −22.99177913516827185211525874014, −22.538909446161570281029022694927, −21.69780317854309326205725582310, −20.73242591141699012829746472655, −20.24377363653251455992909261961, −18.49118070113751126816525934347, −17.96627165422734407951275414880, −16.93817057100427566446693656580, −16.32559334992561120417624364943, −15.206043469195145571906732497851, −14.32912870587186904408420105942, −13.78151041634646189567065046208, −12.79540483454009625190251222714, −12.002020034955540679223176627339, −10.83160926921809028646729243114, −9.639783248788784317420561103965, −8.78805730003782473273471016439, −7.56178219109698327262327027642, −6.65656870444193579277525730385, −5.810322327419620064897686903667, −4.97745968176974217711787531729, −3.73752817254808518757240417459, −2.71469886061783054211514487967, −1.33635037537099742832444258308, 1.016606310658375546173784713102, 1.75870544747129978096433275387, 3.1655149737695426834810288268, 3.95837843028967069427793588086, 5.45152814611402954046446843603, 5.857696831594788515277227075325, 6.97445125576342450126995305137, 8.65297620593284793032524768828, 9.50385133459553852707494334087, 10.33625711186938700495689686126, 11.33698161631429317640403599615, 12.12151427704932354924931995396, 13.23862538614216882017259730432, 13.86200394894640007001906807908, 14.45850093066356147494347501985, 15.710899269448059424829360703280, 16.61294059384698150778487904323, 17.71960795068992831953320742809, 18.61446305746765633363140176479, 19.43246239635149740870331658713, 20.40267395209175860217441252941, 21.28023161700320294211826411266, 21.66661718431928800044288550535, 22.64987109262724270873908331156, 23.47919855393037327856912402182

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