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Properties

Label 1-1441-1441.1072-r1-0-0
Degree $1$
Conductor $1441$
Sign $0.494 + 0.868i$
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  + (0.989 − 0.144i)2-s + (−0.0724 − 0.997i)3-s + (0.958 − 0.285i)4-s + (0.644 + 0.764i)5-s + (−0.215 − 0.976i)6-s + (0.399 + 0.916i)7-s + (0.906 − 0.421i)8-s + (−0.989 + 0.144i)9-s + (0.748 + 0.663i)10-s + (−0.354 − 0.935i)12-s + (0.215 − 0.976i)13-s + (0.527 + 0.849i)14-s + (0.715 − 0.698i)15-s + (0.836 − 0.548i)16-s + (−0.836 + 0.548i)17-s + (−0.958 + 0.285i)18-s + ⋯
L(s)  = 1  + (0.989 − 0.144i)2-s + (−0.0724 − 0.997i)3-s + (0.958 − 0.285i)4-s + (0.644 + 0.764i)5-s + (−0.215 − 0.976i)6-s + (0.399 + 0.916i)7-s + (0.906 − 0.421i)8-s + (−0.989 + 0.144i)9-s + (0.748 + 0.663i)10-s + (−0.354 − 0.935i)12-s + (0.215 − 0.976i)13-s + (0.527 + 0.849i)14-s + (0.715 − 0.698i)15-s + (0.836 − 0.548i)16-s + (−0.836 + 0.548i)17-s + (−0.958 + 0.285i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.494 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.494 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.494 + 0.868i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1072, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ 0.494 + 0.868i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.441096595 + 2.000358335i\)
\(L(\frac12)\) \(\approx\) \(3.441096595 + 2.000358335i\)
\(L(1)\) \(\approx\) \(2.035511116 - 0.07880095463i\)
\(L(1)\) \(\approx\) \(2.035511116 - 0.07880095463i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.989 - 0.144i)T \)
3 \( 1 + (-0.0724 - 0.997i)T \)
5 \( 1 + (0.644 + 0.764i)T \)
7 \( 1 + (0.399 + 0.916i)T \)
13 \( 1 + (0.215 - 0.976i)T \)
17 \( 1 + (-0.836 + 0.548i)T \)
19 \( 1 + (-0.779 + 0.626i)T \)
23 \( 1 + (-0.120 + 0.992i)T \)
29 \( 1 + (0.168 + 0.985i)T \)
31 \( 1 + (0.681 + 0.732i)T \)
37 \( 1 + (-0.958 + 0.285i)T \)
41 \( 1 + (0.779 - 0.626i)T \)
43 \( 1 + (-0.970 - 0.239i)T \)
47 \( 1 + (0.168 - 0.985i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (0.779 + 0.626i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + (0.970 - 0.239i)T \)
71 \( 1 + (-0.485 + 0.873i)T \)
73 \( 1 + (-0.309 + 0.951i)T \)
79 \( 1 + (-0.926 + 0.377i)T \)
83 \( 1 + (0.943 - 0.331i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.215 + 0.976i)T \)
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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

−20.79523337715043407500410309832, −20.09032213318727746119926626117, −19.28300442045465127932704794525, −17.63753738185587406090896444665, −17.205713095223838495105014161100, −16.42777637642265724120826909363, −15.989868171312309957691611866818, −15.02367411722642611421231227144, −14.24320329902023080318863097978, −13.6722456652673489145995750891, −13.047834030239986646484712163286, −11.91671475065303047441516071760, −11.23522585614704589925888137722, −10.54933379065846425874293673803, −9.676015837614511024740553713580, −8.77961971101945010834036484378, −7.9652905873796397082765301394, −6.64292357060740629872602762261, −6.16599500469311914374369935759, −4.92673551457906457725555559807, −4.5282378606202972672400673390, −4.02088466664367097044965491489, −2.69751506726208888395825183459, −1.84264243093024983172229204435, −0.445386106727593322780147291749, 1.39163578641284980186659173202, 2.023117996272334919260303730227, 2.79450848986409253071564855531, 3.591272194383669673630028080631, 5.10862042807514025937116169678, 5.68619146122689385466905062722, 6.360287437864031186621899098540, 7.032344964439897737242416852692, 8.02770708316353816914707859449, 8.83291264267531182636824503361, 10.25110346904104608425722137130, 10.851412014850703230538566443532, 11.64145406163578405672240076086, 12.43263167678667008380895410009, 13.01945003302740298783084750898, 13.77540050398923728360603892798, 14.45276665504442505809673582152, 15.13247446330637861711844172320, 15.77993987322823782790598895083, 17.175534053074230319014133966151, 17.67855252038767340607017666819, 18.50157944890356832945350213328, 19.15216825384608349568997130188, 19.91408493113576217758674032452, 20.78735990215344732491461040801

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