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Properties

Label 1-1441-1441.1055-r1-0-0
Degree $1$
Conductor $1441$
Sign $0.222 - 0.974i$
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  + (0.0724 + 0.997i)2-s + (0.485 + 0.873i)3-s + (−0.989 + 0.144i)4-s + (0.981 − 0.192i)5-s + (−0.836 + 0.548i)6-s + (−0.779 + 0.626i)7-s + (−0.215 − 0.976i)8-s + (−0.527 + 0.849i)9-s + (0.262 + 0.964i)10-s + (−0.607 − 0.794i)12-s + (0.262 − 0.964i)13-s + (−0.681 − 0.732i)14-s + (0.644 + 0.764i)15-s + (0.958 − 0.285i)16-s + (−0.0241 + 0.999i)17-s + (−0.885 − 0.464i)18-s + ⋯
L(s)  = 1  + (0.0724 + 0.997i)2-s + (0.485 + 0.873i)3-s + (−0.989 + 0.144i)4-s + (0.981 − 0.192i)5-s + (−0.836 + 0.548i)6-s + (−0.779 + 0.626i)7-s + (−0.215 − 0.976i)8-s + (−0.527 + 0.849i)9-s + (0.262 + 0.964i)10-s + (−0.607 − 0.794i)12-s + (0.262 − 0.964i)13-s + (−0.681 − 0.732i)14-s + (0.644 + 0.764i)15-s + (0.958 − 0.285i)16-s + (−0.0241 + 0.999i)17-s + (−0.885 − 0.464i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.222 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.222 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.222 - 0.974i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1055, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ 0.222 - 0.974i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.8513573977 + 0.6787439054i\)
\(L(\frac12)\) \(\approx\) \(-0.8513573977 + 0.6787439054i\)
\(L(1)\) \(\approx\) \(0.6343079362 + 0.8977856135i\)
\(L(1)\) \(\approx\) \(0.6343079362 + 0.8977856135i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.0724 + 0.997i)T \)
3 \( 1 + (0.485 + 0.873i)T \)
5 \( 1 + (0.981 - 0.192i)T \)
7 \( 1 + (-0.779 + 0.626i)T \)
13 \( 1 + (0.262 - 0.964i)T \)
17 \( 1 + (-0.0241 + 0.999i)T \)
19 \( 1 + (-0.568 + 0.822i)T \)
23 \( 1 + (0.995 - 0.0965i)T \)
29 \( 1 + (0.527 + 0.849i)T \)
31 \( 1 + (-0.861 + 0.506i)T \)
37 \( 1 + (-0.168 + 0.985i)T \)
41 \( 1 + (-0.958 - 0.285i)T \)
43 \( 1 + (0.906 + 0.421i)T \)
47 \( 1 + (-0.970 - 0.239i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (-0.943 - 0.331i)T \)
61 \( 1 + (0.809 + 0.587i)T \)
67 \( 1 + (-0.906 + 0.421i)T \)
71 \( 1 + (-0.748 - 0.663i)T \)
73 \( 1 + (0.809 - 0.587i)T \)
79 \( 1 + (-0.120 + 0.992i)T \)
83 \( 1 + (-0.715 - 0.698i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (-0.998 + 0.0483i)T \)
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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

−19.88314748481359406958677347351, −19.17207748416700793652547369900, −18.716247578665948470394251012861, −17.85373901639141178523712197187, −17.28003592207960943607023052606, −16.437197927421160752695924845807, −15.03725452675596593649599200925, −14.16589094851756595432606658501, −13.63081139962994732497757916781, −13.18122508688038694952812787024, −12.49820069309330856295464418973, −11.45987109437401030519589611547, −10.82157713700186350614789488754, −9.66834539019352344099611032112, −9.330255272813549580289589203500, −8.56058678321812444285295216383, −7.23163375371844624538959425018, −6.65540773352343509507504386899, −5.73167785544933877523864013572, −4.60088163861776118553262399476, −3.548133333712473172737565494835, −2.720149372663154892322131483790, −2.0686803669503591706641929279, −1.08938476543069141084463093278, −0.1901079813094480850343707726, 1.48337996319349162680746907954, 2.88567888524932317531419229383, 3.50148891838652377772799908993, 4.64793645428037316647812764615, 5.480541851554472172414903446998, 5.9721326132348843185693416249, 6.86167947310579637790148692537, 8.1728955622249231041146092922, 8.71267948711079644317331154253, 9.324566471902872086252557985260, 10.210314453348683090883737900376, 10.61035262158933461028151507050, 12.40063409836574526615049670055, 12.97348275241531955793716400301, 13.627161895084673531688313167009, 14.62812820866138936376206000380, 15.01643537142222418347493517741, 15.81566405791781341758379725861, 16.57140207338226295865736948648, 17.068957222644216711184458818220, 17.96136225995149656039421858656, 18.770654262949358294073164167489, 19.55952957411383555413764078304, 20.5546752786277015746023687592, 21.40043937499031150889770385363

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