Properties

Label 1-1441-1441.299-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.653 - 0.757i$
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  + (−0.861 + 0.506i)2-s + (−0.998 − 0.0483i)3-s + (0.485 − 0.873i)4-s + (0.836 − 0.548i)5-s + (0.885 − 0.464i)6-s + (0.989 − 0.144i)7-s + (0.0241 + 0.999i)8-s + (0.995 + 0.0965i)9-s + (−0.443 + 0.896i)10-s + (−0.527 + 0.849i)12-s + (0.168 − 0.985i)13-s + (−0.779 + 0.626i)14-s + (−0.861 + 0.506i)15-s + (−0.527 − 0.849i)16-s + (−0.970 + 0.239i)17-s + (−0.906 + 0.421i)18-s + ⋯
L(s)  = 1  + (−0.861 + 0.506i)2-s + (−0.998 − 0.0483i)3-s + (0.485 − 0.873i)4-s + (0.836 − 0.548i)5-s + (0.885 − 0.464i)6-s + (0.989 − 0.144i)7-s + (0.0241 + 0.999i)8-s + (0.995 + 0.0965i)9-s + (−0.443 + 0.896i)10-s + (−0.527 + 0.849i)12-s + (0.168 − 0.985i)13-s + (−0.779 + 0.626i)14-s + (−0.861 + 0.506i)15-s + (−0.527 − 0.849i)16-s + (−0.970 + 0.239i)17-s + (−0.906 + 0.421i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.653 - 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.653 - 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.653 - 0.757i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.653 - 0.757i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8225601125 - 0.3768111252i\)
\(L(\frac12)\) \(\approx\) \(0.8225601125 - 0.3768111252i\)
\(L(1)\) \(\approx\) \(0.7013441087 - 0.04121878784i\)
\(L(1)\) \(\approx\) \(0.7013441087 - 0.04121878784i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.861 - 0.506i)T \)
3 \( 1 + (0.998 + 0.0483i)T \)
5 \( 1 + (-0.836 + 0.548i)T \)
7 \( 1 + (-0.989 + 0.144i)T \)
13 \( 1 + (-0.168 + 0.985i)T \)
17 \( 1 + (0.970 - 0.239i)T \)
19 \( 1 + (0.527 - 0.849i)T \)
23 \( 1 + (0.0241 - 0.999i)T \)
29 \( 1 + (-0.215 + 0.976i)T \)
31 \( 1 + (0.0241 + 0.999i)T \)
37 \( 1 + (-0.681 - 0.732i)T \)
41 \( 1 + (-0.970 - 0.239i)T \)
43 \( 1 + (0.779 + 0.626i)T \)
47 \( 1 + (-0.861 + 0.506i)T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (-0.926 - 0.377i)T \)
61 \( 1 + T \)
67 \( 1 + (0.779 - 0.626i)T \)
71 \( 1 + (0.0241 + 0.999i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (-0.836 + 0.548i)T \)
83 \( 1 + (0.681 - 0.732i)T \)
89 \( 1 + (-0.309 + 0.951i)T \)
97 \( 1 + (-0.989 + 0.144i)T \)
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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

−21.00124732747198878482279585562, −20.08743522961313374884724789154, −19.05623702441823100294744014241, −18.2832301242817816582620266643, −17.91489605419719354172338980981, −17.33685045106683036625445942302, −16.58865881942180053951994285449, −15.83816915396339090320945842999, −14.84674227770780464185086989334, −13.915821120767625651553172027959, −12.95710699878738879664897682595, −12.19796913567829879602170897677, −11.18065795772996557473435665058, −10.99334310552900909265225500102, −10.26901245535217389258262382844, −9.18664601372816744708375060572, −8.74079252977630755178463017292, −7.37061820071090303179283709461, −6.77656132981478682925408223359, −6.07969404809852279369421801916, −4.84835041537212902579385387359, −4.19187560276998115438050187489, −2.66436680685320568692285445567, −1.94228756318812041642591392372, −1.07239354676141526053581178227, 0.61036151471226207844068146692, 1.54152167160921266523946520848, 2.23841371954664510883927957226, 4.22261355725140502534758143314, 5.0470816723844250898304361912, 5.82452292622603856085422028579, 6.23489245064462004451945763693, 7.43210669660127173812661096723, 8.09651882499780916108968650865, 8.93921954251755259270531579268, 9.973651600237798642604208982917, 10.41784213316048362230522817572, 11.269668665904312987691427292496, 11.935800002977155998031134254612, 13.10612055330174145273587895971, 13.68207751219938599833928063343, 14.92567617306427386902279673066, 15.4028059140788368151887523793, 16.443092071885643470601875774430, 17.02448459554490019466565353534, 17.64307788291407640777275261315, 17.95626899018727819038911607754, 18.78090117295460537241197767286, 19.83844956779473319737717088414, 20.678542196961247561901034160768

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