Properties

Label 1-1441-1441.1209-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: $\chi_{1441} (1209, \cdot )$
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

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

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