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Properties

Label 1-1441-1441.1000-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.952 - 0.303i$
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 + (0.836 + 0.548i)3-s + (0.485 + 0.873i)4-s + (0.779 − 0.626i)5-s + (−0.443 − 0.896i)6-s + (−0.715 − 0.698i)7-s + (0.0241 − 0.999i)8-s + (0.399 + 0.916i)9-s + (−0.989 + 0.144i)10-s + (−0.0724 + 0.997i)12-s + (0.989 + 0.144i)13-s + (0.262 + 0.964i)14-s + (0.995 − 0.0965i)15-s + (−0.527 + 0.849i)16-s + (0.644 + 0.764i)17-s + (0.120 − 0.992i)18-s + ⋯
L(s)  = 1  + (−0.861 − 0.506i)2-s + (0.836 + 0.548i)3-s + (0.485 + 0.873i)4-s + (0.779 − 0.626i)5-s + (−0.443 − 0.896i)6-s + (−0.715 − 0.698i)7-s + (0.0241 − 0.999i)8-s + (0.399 + 0.916i)9-s + (−0.989 + 0.144i)10-s + (−0.0724 + 0.997i)12-s + (0.989 + 0.144i)13-s + (0.262 + 0.964i)14-s + (0.995 − 0.0965i)15-s + (−0.527 + 0.849i)16-s + (0.644 + 0.764i)17-s + (0.120 − 0.992i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.952 - 0.303i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1000, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.952 - 0.303i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.634814526 - 0.2536643444i\)
\(L(\frac12)\) \(\approx\) \(1.634814526 - 0.2536643444i\)
\(L(1)\) \(\approx\) \(1.103810351 - 0.1378061645i\)
\(L(1)\) \(\approx\) \(1.103810351 - 0.1378061645i\)

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 \( 1 + (0.836 + 0.548i)T \)
5 \( 1 + (0.779 - 0.626i)T \)
7 \( 1 + (-0.715 - 0.698i)T \)
13 \( 1 + (0.989 + 0.144i)T \)
17 \( 1 + (0.644 + 0.764i)T \)
19 \( 1 + (-0.970 + 0.239i)T \)
23 \( 1 + (0.943 - 0.331i)T \)
29 \( 1 + (0.399 - 0.916i)T \)
31 \( 1 + (-0.958 + 0.285i)T \)
37 \( 1 + (-0.981 + 0.192i)T \)
41 \( 1 + (0.527 + 0.849i)T \)
43 \( 1 + (0.998 - 0.0483i)T \)
47 \( 1 + (0.748 - 0.663i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (0.926 - 0.377i)T \)
61 \( 1 + (0.809 + 0.587i)T \)
67 \( 1 + (0.998 + 0.0483i)T \)
71 \( 1 + (-0.568 - 0.822i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (-0.354 - 0.935i)T \)
83 \( 1 + (-0.906 + 0.421i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (0.168 + 0.985i)T \)
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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

−20.77337847323273247296398542783, −19.68033492394215962236761707472, −18.93918837109635526406947120917, −18.66674400344363657183205828344, −17.90655371959486264996962113438, −17.24420425849633825560133600258, −16.09327378671679595666167107474, −15.55049663414119419522099649025, −14.65453913421246074311996208552, −14.18189937313383392778814701242, −13.24674227588690944121521657645, −12.56354400967709349299954166006, −11.34287487977165247909932013316, −10.5133938420443987989908040183, −9.66894777483095928102565002464, −8.98308819447476234171257492066, −8.5738382365165294253772073082, −7.2917658435049603282191849724, −6.878898349867559213465697469474, −6.017712215716080582731806039610, −5.38052487068196215332223456169, −3.57029802801485852993104932515, −2.7131961190181995506484958891, −2.03140758606554065741394258529, −0.96741800405441137830763741489, 0.98141221498863475553023407343, 1.86167945448532908106418831859, 2.82279583362445669860518984275, 3.757536178229388847512928167045, 4.33980375216835919689562650792, 5.78484377690939382204678229918, 6.70939955065090377566583758665, 7.69811777432647434177583089459, 8.68033371356567847431194327902, 8.929146620380252618455732766048, 9.97455217746955250534753298010, 10.34162239699065640067036099624, 11.07768514211672836281186310347, 12.49644044462579430734506467919, 13.0300756693563353127639897591, 13.64390398609068613338839505129, 14.60305307962343483349432334053, 15.681113381633809074202554522660, 16.35299860706860555301497035377, 16.90625344625374333000590790407, 17.54751000179271499264107646782, 18.792544855727850940637591436638, 19.15630529218119893865196943092, 20.036182409320540888673701741820, 20.6372369223182367032658211611

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