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Properties

Label 1-1441-1441.1092-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.998 + 0.0528i$
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  + (0.0241 + 0.999i)2-s + (−0.989 − 0.144i)3-s + (−0.998 + 0.0483i)4-s + (−0.168 − 0.985i)5-s + (0.120 − 0.992i)6-s + (−0.906 + 0.421i)7-s + (−0.0724 − 0.997i)8-s + (0.958 + 0.285i)9-s + (0.981 − 0.192i)10-s + (0.995 + 0.0965i)12-s + (0.485 − 0.873i)13-s + (−0.443 − 0.896i)14-s + (0.0241 + 0.999i)15-s + (0.995 − 0.0965i)16-s + (−0.748 + 0.663i)17-s + (−0.262 + 0.964i)18-s + ⋯
L(s)  = 1  + (0.0241 + 0.999i)2-s + (−0.989 − 0.144i)3-s + (−0.998 + 0.0483i)4-s + (−0.168 − 0.985i)5-s + (0.120 − 0.992i)6-s + (−0.906 + 0.421i)7-s + (−0.0724 − 0.997i)8-s + (0.958 + 0.285i)9-s + (0.981 − 0.192i)10-s + (0.995 + 0.0965i)12-s + (0.485 − 0.873i)13-s + (−0.443 − 0.896i)14-s + (0.0241 + 0.999i)15-s + (0.995 − 0.0965i)16-s + (−0.748 + 0.663i)17-s + (−0.262 + 0.964i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0528i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0528i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.998 + 0.0528i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1092, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.998 + 0.0528i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6197313628 + 0.01639732845i\)
\(L(\frac12)\) \(\approx\) \(0.6197313628 + 0.01639732845i\)
\(L(1)\) \(\approx\) \(0.5873593106 + 0.1622251993i\)
\(L(1)\) \(\approx\) \(0.5873593106 + 0.1622251993i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.0241 + 0.999i)T \)
3 \( 1 + (-0.989 - 0.144i)T \)
5 \( 1 + (-0.168 - 0.985i)T \)
7 \( 1 + (-0.906 + 0.421i)T \)
13 \( 1 + (0.485 - 0.873i)T \)
17 \( 1 + (-0.748 + 0.663i)T \)
19 \( 1 + (0.995 + 0.0965i)T \)
23 \( 1 + (-0.0724 + 0.997i)T \)
29 \( 1 + (-0.607 + 0.794i)T \)
31 \( 1 + (-0.0724 - 0.997i)T \)
37 \( 1 + (0.779 - 0.626i)T \)
41 \( 1 + (-0.748 - 0.663i)T \)
43 \( 1 + (-0.443 + 0.896i)T \)
47 \( 1 + (0.0241 + 0.999i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (0.399 + 0.916i)T \)
61 \( 1 + T \)
67 \( 1 + (-0.443 - 0.896i)T \)
71 \( 1 + (-0.0724 - 0.997i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (-0.168 - 0.985i)T \)
83 \( 1 + (0.779 + 0.626i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (-0.906 + 0.421i)T \)
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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

−20.66880054034092985812928071702, −20.03558071649590050525555695687, −19.006884772574243228114164119020, −18.57620829458807610612483359101, −17.98298388228779998691275012407, −17.08240571164153145783375019383, −16.26682974188615223875016319824, −15.56645453244280451584457020485, −14.46022732660657141042841975347, −13.62707096102000666572885689226, −13.05329350638663183368289341013, −11.98955420110206560185253756748, −11.44377435438295260857171655215, −10.89494471344901904129954943094, −9.94350930781498636877389930400, −9.678973743876742636288723912316, −8.457110307483176481493365575, −7.06011143227731913726024934199, −6.65367089333559810521825466648, −5.64183159405950613776248080863, −4.57906246149582099395574256894, −3.83412433069842647943141063901, −3.06418398035802836276183262940, −1.989785846036269000104213852251, −0.69390593195930845554828083693, 0.45372290141664412261272051586, 1.52753018332094641417824031809, 3.38088719233037549437095395359, 4.21478682925221473845077923621, 5.21831797326278489621701687585, 5.74674826297837318485641913495, 6.34173366319503060379856548160, 7.42096276976091379231744530206, 8.052121310735619233590339315323, 9.17136035194960698370499206315, 9.60524798093651512578449164666, 10.67954028776986472056852120591, 11.741777445679701476332856003586, 12.596796241347174872025034205690, 13.07559348604219316071097517555, 13.6011447375759838982463951426, 15.14460972212023970057087074663, 15.60426567011393736516294633740, 16.298857283225308826675195871662, 16.74189898983899909591706185074, 17.689013725012755288690455128872, 18.09667633168182853622216156441, 19.06777745593925201541276791584, 19.80703234000719915051810978174, 20.850405998758038518805312944704

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