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Properties

Label 1-1441-1441.1044-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.997 - 0.0718i$
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  + (0.995 + 0.0965i)2-s + (0.779 + 0.626i)3-s + (0.981 + 0.192i)4-s + (−0.262 − 0.964i)5-s + (0.715 + 0.698i)6-s + (0.989 − 0.144i)7-s + (0.958 + 0.285i)8-s + (0.215 + 0.976i)9-s + (−0.168 − 0.985i)10-s + (0.644 + 0.764i)12-s + (0.168 − 0.985i)13-s + (0.998 − 0.0483i)14-s + (0.399 − 0.916i)15-s + (0.926 + 0.377i)16-s + (−0.527 − 0.849i)17-s + (0.120 + 0.992i)18-s + ⋯
L(s)  = 1  + (0.995 + 0.0965i)2-s + (0.779 + 0.626i)3-s + (0.981 + 0.192i)4-s + (−0.262 − 0.964i)5-s + (0.715 + 0.698i)6-s + (0.989 − 0.144i)7-s + (0.958 + 0.285i)8-s + (0.215 + 0.976i)9-s + (−0.168 − 0.985i)10-s + (0.644 + 0.764i)12-s + (0.168 − 0.985i)13-s + (0.998 − 0.0483i)14-s + (0.399 − 0.916i)15-s + (0.926 + 0.377i)16-s + (−0.527 − 0.849i)17-s + (0.120 + 0.992i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0718i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.997 - 0.0718i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1044, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.997 - 0.0718i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.434806601 - 0.1596022943i\)
\(L(\frac12)\) \(\approx\) \(4.434806601 - 0.1596022943i\)
\(L(1)\) \(\approx\) \(2.645920529 + 0.09887373169i\)
\(L(1)\) \(\approx\) \(2.645920529 + 0.09887373169i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.995 + 0.0965i)T \)
3 \( 1 + (0.779 + 0.626i)T \)
5 \( 1 + (-0.262 - 0.964i)T \)
7 \( 1 + (0.989 - 0.144i)T \)
13 \( 1 + (0.168 - 0.985i)T \)
17 \( 1 + (-0.527 - 0.849i)T \)
19 \( 1 + (-0.970 - 0.239i)T \)
23 \( 1 + (0.607 - 0.794i)T \)
29 \( 1 + (0.215 - 0.976i)T \)
31 \( 1 + (0.943 - 0.331i)T \)
37 \( 1 + (0.681 + 0.732i)T \)
41 \( 1 + (-0.926 + 0.377i)T \)
43 \( 1 + (-0.836 + 0.548i)T \)
47 \( 1 + (0.748 + 0.663i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (-0.0724 + 0.997i)T \)
61 \( 1 + (-0.309 - 0.951i)T \)
67 \( 1 + (-0.836 - 0.548i)T \)
71 \( 1 + (-0.568 + 0.822i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (-0.354 + 0.935i)T \)
83 \( 1 + (0.485 + 0.873i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (0.443 + 0.896i)T \)
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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

−20.89199489038616824203361510911, −19.98405778751659712975647552573, −19.258028802271804115725163873252, −18.80241529002221910606614975406, −17.846717031090913574262556073677, −17.025307282246045275186704155837, −15.75898195591873354090972223377, −14.99935977024207345238290359212, −14.68042112055051946863537384148, −13.93508024229402353194875114897, −13.3427917939975064763356762465, −12.349151823333792773915365788473, −11.67134599853410146567410473301, −10.98137679735635575158401579157, −10.20947358990815724540949846667, −8.847943090695295537544165903513, −8.12778974513700173222866223492, −7.18064699057418669587699281147, −6.70046142917177150045990603323, −5.854667191629282543106563794755, −4.5747759168132992722311216512, −3.86269720967984321929106643657, −3.03391482383923807915534425670, −2.03538746328326876176982944670, −1.59486551763441746553890633580, 1.155078563528952850894046120880, 2.30855093834210090031152373125, 3.04784983702323458144804301588, 4.31132083579361278980539027392, 4.57521561714097008589953581445, 5.26622717490161266047586818541, 6.41248617550239427400397547269, 7.69615438533170275695840294298, 8.12876711925086531118899399401, 8.85964907575165124723087690948, 10.01451239483082336620002211336, 10.878476867827453148131493950063, 11.54944492805237764186910524260, 12.48994511488646415090037715744, 13.37562792064029193809341137763, 13.71203848665958891236838637032, 14.8428902416863220501179299011, 15.243467525286189857196716832837, 15.87143431843845652651619702775, 16.82507961508592540127897676556, 17.31105202825573782888108471, 18.64618766222099552730167213316, 19.72819160921331846967379257573, 20.22634203427035089667579954626, 20.8459361477036106433460587878

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