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Properties

Label 1-1441-1441.405-r0-0-0
Degree $1$
Conductor $1441$
Sign $-0.316 - 0.948i$
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  + (0.485 + 0.873i)2-s + (0.215 − 0.976i)3-s + (−0.527 + 0.849i)4-s + (−0.861 − 0.506i)5-s + (0.958 − 0.285i)6-s + (0.568 − 0.822i)7-s + (−0.998 − 0.0483i)8-s + (−0.906 − 0.421i)9-s + (0.0241 − 0.999i)10-s + (0.715 + 0.698i)12-s + (0.568 − 0.822i)13-s + (0.995 + 0.0965i)14-s + (−0.681 + 0.732i)15-s + (−0.443 − 0.896i)16-s + (0.715 − 0.698i)17-s + (−0.0724 − 0.997i)18-s + ⋯
L(s)  = 1  + (0.485 + 0.873i)2-s + (0.215 − 0.976i)3-s + (−0.527 + 0.849i)4-s + (−0.861 − 0.506i)5-s + (0.958 − 0.285i)6-s + (0.568 − 0.822i)7-s + (−0.998 − 0.0483i)8-s + (−0.906 − 0.421i)9-s + (0.0241 − 0.999i)10-s + (0.715 + 0.698i)12-s + (0.568 − 0.822i)13-s + (0.995 + 0.0965i)14-s + (−0.681 + 0.732i)15-s + (−0.443 − 0.896i)16-s + (0.715 − 0.698i)17-s + (−0.0724 − 0.997i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.316 - 0.948i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (405, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ -0.316 - 0.948i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7655872871 - 1.062914437i\)
\(L(\frac12)\) \(\approx\) \(0.7655872871 - 1.062914437i\)
\(L(1)\) \(\approx\) \(1.111132972 - 0.1977611285i\)
\(L(1)\) \(\approx\) \(1.111132972 - 0.1977611285i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.485 + 0.873i)T \)
3 \( 1 + (0.215 - 0.976i)T \)
5 \( 1 + (-0.861 - 0.506i)T \)
7 \( 1 + (0.568 - 0.822i)T \)
13 \( 1 + (0.568 - 0.822i)T \)
17 \( 1 + (0.715 - 0.698i)T \)
19 \( 1 + (-0.168 - 0.985i)T \)
23 \( 1 + (0.836 + 0.548i)T \)
29 \( 1 + (0.120 + 0.992i)T \)
31 \( 1 + (0.779 + 0.626i)T \)
37 \( 1 + (-0.970 - 0.239i)T \)
41 \( 1 + (-0.989 - 0.144i)T \)
43 \( 1 + (-0.861 - 0.506i)T \)
47 \( 1 + (-0.906 - 0.421i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (0.715 - 0.698i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (-0.861 + 0.506i)T \)
71 \( 1 + (-0.262 - 0.964i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (0.995 - 0.0965i)T \)
83 \( 1 + (-0.527 - 0.849i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (-0.607 - 0.794i)T \)
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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

−21.05333109895348607046173462987, −20.51394604445826653412500959690, −19.304503210085878131954373056597, −19.05523372671580019617155593336, −18.31615203179768253700077286982, −17.154128916417028410339567270623, −16.19839610785540439231406954161, −15.35407593928548868999252473822, −14.80190919079550809793288491801, −14.376555490216032701430949537298, −13.397336419047166387705243880769, −12.19527350194349564126953080866, −11.686656471235225311481223495145, −11.087159780641053678664952775673, −10.298137736665999422583402574260, −9.602419195947436223608244541794, −8.42831773630796469147685990950, −8.26093277740158779006910008769, −6.53529278960374469597112896941, −5.70350594230898536258639673487, −4.758278053355619437960868335371, −4.0856775325024672833904244455, −3.36248500934928387059466780483, −2.56394318016740357501400600448, −1.50002166019950639799598398968, 0.43387921142355149856997695129, 1.39389925184728662533493478184, 3.13782489178042773474561109626, 3.52595371157295889775900465289, 4.92612641264925737181782636244, 5.22628541808567836176368792734, 6.66947343074951294320002675889, 7.140070114859443553622138603651, 7.91416328594793206570186248377, 8.41550603202135753782209817635, 9.18833672780823092574709545619, 10.71231651479459820674594824245, 11.63396455534990487070596616149, 12.228741537615087840548980693280, 13.15063548816000253911335220340, 13.54380398616229273982324981052, 14.40057030294692483590355862171, 15.15688317892860896106204386091, 15.8918199347839026609141387933, 16.77215606776456272298525830902, 17.43183777027595516483415729060, 18.03770987237183497102519965746, 18.93004269388241993568264588339, 19.79607602648767749427074971817, 20.52005388007561967681026304106

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