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Properties

Label 1-1441-1441.103-r1-0-0
Degree $1$
Conductor $1441$
Sign $-0.378 + 0.925i$
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  + (−0.958 + 0.285i)2-s + (0.715 − 0.698i)3-s + (0.836 − 0.548i)4-s + (−0.989 + 0.144i)5-s + (−0.485 + 0.873i)6-s + (0.120 − 0.992i)7-s + (−0.644 + 0.764i)8-s + (0.0241 − 0.999i)9-s + (0.906 − 0.421i)10-s + (0.215 − 0.976i)12-s + (0.120 − 0.992i)13-s + (0.168 + 0.985i)14-s + (−0.607 + 0.794i)15-s + (0.399 − 0.916i)16-s + (−0.215 − 0.976i)17-s + (0.262 + 0.964i)18-s + ⋯
L(s)  = 1  + (−0.958 + 0.285i)2-s + (0.715 − 0.698i)3-s + (0.836 − 0.548i)4-s + (−0.989 + 0.144i)5-s + (−0.485 + 0.873i)6-s + (0.120 − 0.992i)7-s + (−0.644 + 0.764i)8-s + (0.0241 − 0.999i)9-s + (0.906 − 0.421i)10-s + (0.215 − 0.976i)12-s + (0.120 − 0.992i)13-s + (0.168 + 0.985i)14-s + (−0.607 + 0.794i)15-s + (0.399 − 0.916i)16-s + (−0.215 − 0.976i)17-s + (0.262 + 0.964i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.378 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.378 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.378 + 0.925i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ -0.378 + 0.925i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3358811608 - 0.5004189661i\)
\(L(\frac12)\) \(\approx\) \(-0.3358811608 - 0.5004189661i\)
\(L(1)\) \(\approx\) \(0.5681787315 - 0.3339635053i\)
\(L(1)\) \(\approx\) \(0.5681787315 - 0.3339635053i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.958 + 0.285i)T \)
3 \( 1 + (0.715 - 0.698i)T \)
5 \( 1 + (-0.989 + 0.144i)T \)
7 \( 1 + (0.120 - 0.992i)T \)
13 \( 1 + (0.120 - 0.992i)T \)
17 \( 1 + (-0.215 - 0.976i)T \)
19 \( 1 + (-0.995 + 0.0965i)T \)
23 \( 1 + (0.527 - 0.849i)T \)
29 \( 1 + (-0.568 + 0.822i)T \)
31 \( 1 + (-0.926 - 0.377i)T \)
37 \( 1 + (0.354 - 0.935i)T \)
41 \( 1 + (-0.861 - 0.506i)T \)
43 \( 1 + (-0.989 + 0.144i)T \)
47 \( 1 + (-0.0241 + 0.999i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (0.215 + 0.976i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + (0.989 + 0.144i)T \)
71 \( 1 + (0.0724 - 0.997i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (0.168 - 0.985i)T \)
83 \( 1 + (-0.836 - 0.548i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (0.681 - 0.732i)T \)
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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

−21.05286587980698940180182648082, −20.0309020818945214576658860925, −19.60079642402083134359461564072, −18.835618936341083052040480706723, −18.48819312408200105861291936198, −16.97609996324262134858800598552, −16.7188093531184590057211883149, −15.56596742732503309816604998714, −15.31660094814986171201709860555, −14.67315340312275107260484520312, −13.28485969675647882418795338631, −12.49310083188494688831874232582, −11.49796310751082775668303348962, −11.16611790110866425638482102551, −10.12413322991993430693283901775, −9.27547215923314666633575588047, −8.6045182452482759265017940011, −8.259398410212439063057075375837, −7.28130582782525619690479388242, −6.318047309854466519026232670050, −5.0266965350557323631781092998, −3.98939883309325986684893583934, −3.37996406863066685877936119932, −2.29862480173302653978615492978, −1.57888746136140649367520785971, 0.200305840310021824544886344114, 0.68578558764845037047611495983, 1.84977243910443567329931773937, 2.9295424675112326621791682362, 3.68583662559570188206912325350, 4.89268358329126187713429305220, 6.235981051752356948532365021471, 7.175973853617776144240768989542, 7.3889643429619549923623593229, 8.300998727886182160462839619739, 8.81074861422563958121358288827, 9.87245263961704881698275769537, 10.805100539198343418168789738996, 11.29232208174258797946798396730, 12.422431246038607515701908493, 13.05965982222632269336268489333, 14.20988936210354643750319044925, 14.810291377034526621903254806712, 15.428619632152802862996703012988, 16.37926873810764964672605079366, 17.01243836968773939515382467082, 18.05320973889933347495188556413, 18.44782118283889695366533281806, 19.29980873890573238727309934704, 19.94876379183631887248680152573

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