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Properties

Label 1-1441-1441.652-r1-0-0
Degree $1$
Conductor $1441$
Sign $0.996 - 0.0799i$
Analytic cond. $154.856$
Root an. cond. $154.856$
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  + (0.607 − 0.794i)2-s + (−0.168 + 0.985i)3-s + (−0.262 − 0.964i)4-s + (−0.443 − 0.896i)5-s + (0.681 + 0.732i)6-s + (0.120 − 0.992i)7-s + (−0.926 − 0.377i)8-s + (−0.943 − 0.331i)9-s + (−0.981 − 0.192i)10-s + (0.995 − 0.0965i)12-s + (0.120 − 0.992i)13-s + (−0.715 − 0.698i)14-s + (0.958 − 0.285i)15-s + (−0.861 + 0.506i)16-s + (−0.995 − 0.0965i)17-s + (−0.836 + 0.548i)18-s + ⋯
L(s)  = 1  + (0.607 − 0.794i)2-s + (−0.168 + 0.985i)3-s + (−0.262 − 0.964i)4-s + (−0.443 − 0.896i)5-s + (0.681 + 0.732i)6-s + (0.120 − 0.992i)7-s + (−0.926 − 0.377i)8-s + (−0.943 − 0.331i)9-s + (−0.981 − 0.192i)10-s + (0.995 − 0.0965i)12-s + (0.120 − 0.992i)13-s + (−0.715 − 0.698i)14-s + (0.958 − 0.285i)15-s + (−0.861 + 0.506i)16-s + (−0.995 − 0.0965i)17-s + (−0.836 + 0.548i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.996 - 0.0799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.996 - 0.0799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.996 - 0.0799i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (652, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ 0.996 - 0.0799i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.067272788 - 0.04272402436i\)
\(L(\frac12)\) \(\approx\) \(1.067272788 - 0.04272402436i\)
\(L(1)\) \(\approx\) \(0.8882289586 - 0.4286245030i\)
\(L(1)\) \(\approx\) \(0.8882289586 - 0.4286245030i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.607 - 0.794i)T \)
3 \( 1 + (-0.168 + 0.985i)T \)
5 \( 1 + (-0.443 - 0.896i)T \)
7 \( 1 + (0.120 - 0.992i)T \)
13 \( 1 + (0.120 - 0.992i)T \)
17 \( 1 + (-0.995 - 0.0965i)T \)
19 \( 1 + (-0.215 + 0.976i)T \)
23 \( 1 + (0.0724 + 0.997i)T \)
29 \( 1 + (-0.568 + 0.822i)T \)
31 \( 1 + (-0.644 + 0.764i)T \)
37 \( 1 + (0.354 - 0.935i)T \)
41 \( 1 + (0.399 + 0.916i)T \)
43 \( 1 + (-0.443 - 0.896i)T \)
47 \( 1 + (0.943 + 0.331i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (0.995 + 0.0965i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + (0.443 - 0.896i)T \)
71 \( 1 + (0.527 + 0.849i)T \)
73 \( 1 + (0.809 - 0.587i)T \)
79 \( 1 + (-0.715 + 0.698i)T \)
83 \( 1 + (0.262 - 0.964i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (-0.485 - 0.873i)T \)
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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

−20.63845420049973874168716333125, −19.56891547412977484795848680298, −18.75531179867052321328886750389, −18.339656838683783248667749986081, −17.586486970907082920807188617343, −16.78670158391888442337593989676, −15.85152000268326839203014164413, −15.08824624403103092789553869989, −14.58817488564525834900691240720, −13.70371270418567640021345122537, −13.06091049360211166539839138847, −12.20592844945124348824320102343, −11.47473230232841548523581999739, −11.083774039115202887730738795823, −9.349683029064236554342100571248, −8.573940302564313445269391670701, −7.90195771160322920314655790936, −6.877385244894997186372397231015, −6.580479071510840278904676215875, −5.78595762605333471556469733010, −4.74763056124975089125170048379, −3.809714383712457479725738899853, −2.50649924505458909704732373633, −2.266060852455706146686104395983, −0.24306875103674177862651407160, 0.65553078655186031192264542833, 1.65430131526856626617029677043, 3.11035876992891663350675343651, 3.86377290042842345587589753348, 4.34853239144778290360372835313, 5.27121536770504648315329906125, 5.80097908466298796287430532178, 7.19313739174544787746477788376, 8.31359727490986884295356511668, 9.132812746934300958531792999522, 9.867238586100780547498019319568, 10.78891451840141836737895214524, 11.07390090193857190838209176414, 12.116829643262833658643632081630, 12.83384670865279651040901724573, 13.58241532833863321847375162264, 14.42229175086164879237704437280, 15.23779690112219085788380402979, 15.87015260228710029523157107235, 16.65797360112181175545994141, 17.44804027538295068557562272309, 18.25738241713479261871801671106, 19.65714105095262977440123549712, 19.98288651843002836862835596250, 20.488112602796434040941500026878

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