Properties

Label 1-1441-1441.1038-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.878 + 0.478i$
Analytic cond. $6.69197$
Root an. cond. $6.69197$
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  + (0.926 + 0.377i)2-s + (−0.681 − 0.732i)3-s + (0.715 + 0.698i)4-s + (−0.906 + 0.421i)5-s + (−0.354 − 0.935i)6-s + (0.779 + 0.626i)7-s + (0.399 + 0.916i)8-s + (−0.0724 + 0.997i)9-s + (−0.998 + 0.0483i)10-s + (0.0241 − 0.999i)12-s + (−0.262 − 0.964i)13-s + (0.485 + 0.873i)14-s + (0.926 + 0.377i)15-s + (0.0241 + 0.999i)16-s + (0.568 − 0.822i)17-s + (−0.443 + 0.896i)18-s + ⋯
L(s)  = 1  + (0.926 + 0.377i)2-s + (−0.681 − 0.732i)3-s + (0.715 + 0.698i)4-s + (−0.906 + 0.421i)5-s + (−0.354 − 0.935i)6-s + (0.779 + 0.626i)7-s + (0.399 + 0.916i)8-s + (−0.0724 + 0.997i)9-s + (−0.998 + 0.0483i)10-s + (0.0241 − 0.999i)12-s + (−0.262 − 0.964i)13-s + (0.485 + 0.873i)14-s + (0.926 + 0.377i)15-s + (0.0241 + 0.999i)16-s + (0.568 − 0.822i)17-s + (−0.443 + 0.896i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.878 + 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.878 + 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.878 + 0.478i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1038, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.878 + 0.478i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.039429525 + 0.5193793622i\)
\(L(\frac12)\) \(\approx\) \(2.039429525 + 0.5193793622i\)
\(L(1)\) \(\approx\) \(1.426032548 + 0.2301057396i\)
\(L(1)\) \(\approx\) \(1.426032548 + 0.2301057396i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.926 + 0.377i)T \)
3 \( 1 + (-0.681 - 0.732i)T \)
5 \( 1 + (-0.906 + 0.421i)T \)
7 \( 1 + (0.779 + 0.626i)T \)
13 \( 1 + (-0.262 - 0.964i)T \)
17 \( 1 + (0.568 - 0.822i)T \)
19 \( 1 + (0.0241 - 0.999i)T \)
23 \( 1 + (0.399 - 0.916i)T \)
29 \( 1 + (-0.527 + 0.849i)T \)
31 \( 1 + (0.399 + 0.916i)T \)
37 \( 1 + (-0.168 - 0.985i)T \)
41 \( 1 + (0.568 + 0.822i)T \)
43 \( 1 + (0.485 - 0.873i)T \)
47 \( 1 + (0.926 + 0.377i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.958 + 0.285i)T \)
61 \( 1 + T \)
67 \( 1 + (0.485 + 0.873i)T \)
71 \( 1 + (0.399 + 0.916i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (-0.906 + 0.421i)T \)
83 \( 1 + (-0.168 + 0.985i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (0.779 + 0.626i)T \)
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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

−20.95586627666435843672171832468, −20.24540426585994876245944235781, −19.280008261280194869318515175549, −18.7163832382441538254867024107, −17.22718020846608739482483040429, −16.84887811608359686963724060524, −16.043130813109853890068917181838, −15.24991603861090305009475078563, −14.70169951920078192239440261172, −13.91987329306471958492322637966, −12.864967567240620895288931168444, −12.018148247154895526277554175043, −11.56683446141118295287854778635, −10.95008694101083463488607851400, −10.12538225706486622181431133320, −9.312924971210881019386165586266, −8.02038939537302701075673432326, −7.27783941326463140112506566162, −6.19493780035996793468392982613, −5.38553436095733613561602140184, −4.539661617458968662444327415857, −4.03725548371549820627314481996, −3.436681813990269279162465364879, −1.8290245137216430383393545762, −0.87940264407942825077472359253, 0.91360389199850296356099937748, 2.4207143327140033458441350779, 2.9274365137576205121988457256, 4.24443758581332645423497671439, 5.156671356503138733496944905422, 5.53091665077033530753789947384, 6.792387551626059095559987173, 7.23732677643821974319278611368, 8.00708033550238815027587209849, 8.705660669984516872583408441153, 10.48198464674537748522276798414, 11.206428505217539561519068889410, 11.64375747082710996972567537467, 12.56109143629928807694526885567, 12.82517448841639816898418249927, 14.21441494184309978451875031548, 14.50444570419157912139257620609, 15.605402376872694769208634724206, 15.96364569902392321821152073672, 17.009431604309376901140295566732, 17.77972204271370382736069961249, 18.37789241637425434457485354788, 19.284328935233551407612603207047, 20.091236396321885672409360077890, 20.87501325799833568394788562819

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