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Properties

Label 1-1441-1441.546-r0-0-0
Degree $1$
Conductor $1441$
Sign $-0.916 + 0.400i$
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  + (0.644 + 0.764i)2-s + (0.485 + 0.873i)3-s + (−0.168 + 0.985i)4-s + (0.981 − 0.192i)5-s + (−0.354 + 0.935i)6-s + (0.998 − 0.0483i)7-s + (−0.861 + 0.506i)8-s + (−0.527 + 0.849i)9-s + (0.779 + 0.626i)10-s + (−0.943 + 0.331i)12-s + (−0.836 − 0.548i)13-s + (0.681 + 0.732i)14-s + (0.644 + 0.764i)15-s + (−0.943 − 0.331i)16-s + (0.568 + 0.822i)17-s + (−0.989 + 0.144i)18-s + ⋯
L(s)  = 1  + (0.644 + 0.764i)2-s + (0.485 + 0.873i)3-s + (−0.168 + 0.985i)4-s + (0.981 − 0.192i)5-s + (−0.354 + 0.935i)6-s + (0.998 − 0.0483i)7-s + (−0.861 + 0.506i)8-s + (−0.527 + 0.849i)9-s + (0.779 + 0.626i)10-s + (−0.943 + 0.331i)12-s + (−0.836 − 0.548i)13-s + (0.681 + 0.732i)14-s + (0.644 + 0.764i)15-s + (−0.943 − 0.331i)16-s + (0.568 + 0.822i)17-s + (−0.989 + 0.144i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.916 + 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.916 + 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.916 + 0.400i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (546, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ -0.916 + 0.400i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6467085031 + 3.093837207i\)
\(L(\frac12)\) \(\approx\) \(0.6467085031 + 3.093837207i\)
\(L(1)\) \(\approx\) \(1.329109265 + 1.497068807i\)
\(L(1)\) \(\approx\) \(1.329109265 + 1.497068807i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.644 + 0.764i)T \)
3 \( 1 + (0.485 + 0.873i)T \)
5 \( 1 + (0.981 - 0.192i)T \)
7 \( 1 + (0.998 - 0.0483i)T \)
13 \( 1 + (-0.836 - 0.548i)T \)
17 \( 1 + (0.568 + 0.822i)T \)
19 \( 1 + (-0.943 + 0.331i)T \)
23 \( 1 + (0.861 + 0.506i)T \)
29 \( 1 + (-0.0724 + 0.997i)T \)
31 \( 1 + (0.861 - 0.506i)T \)
37 \( 1 + (-0.715 + 0.698i)T \)
41 \( 1 + (-0.568 + 0.822i)T \)
43 \( 1 + (0.681 - 0.732i)T \)
47 \( 1 + (-0.644 - 0.764i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (-0.607 + 0.794i)T \)
61 \( 1 - T \)
67 \( 1 + (0.681 + 0.732i)T \)
71 \( 1 + (0.861 - 0.506i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (0.981 - 0.192i)T \)
83 \( 1 + (0.715 + 0.698i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
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

−20.75359677044347772939047483289, −19.5869116118474027047047429660, −18.98510110803374872809238112037, −18.3742707866950564106818442897, −17.541304191916543391657648300644, −17.0377679786258933309593202644, −15.41681662835190498747322765922, −14.65020683263471327616311010230, −14.09731497850246884002549610311, −13.723806631696216273693120351630, −12.69492657043525704205092670323, −12.18959890361584521474906257541, −11.29197478899493690658471697701, −10.5626245139417522138193270226, −9.488196317606824484844926161032, −9.005530245211976035364863224319, −7.876440500114381971847275877414, −6.856821746196621148518844514674, −6.208987769112903766041749393468, −5.1599853801904506740155690971, −4.55054235812004625165326324349, −3.16547871350613436333817227159, −2.36274726782304127372581666005, −1.87491166969628547396974275542, −0.86709945417925250130928077902, 1.6910935991052708734996655024, 2.650273012012650178513010598876, 3.554944039456097365080079264469, 4.6303944535972551206904347000, 5.116473789689398062391609790563, 5.78443389548410100653798769093, 6.87363917463134476827315269294, 7.99668715588437667233860368408, 8.43108879648069886984319616280, 9.32224726455958078579701294766, 10.21595836498429252906862148560, 10.93283109919495596059908453123, 12.07744840018498955890474367090, 12.92413986677651284907113743492, 13.69635820413181330123847179075, 14.4210920357052086425712616009, 14.9195872570485301189756412150, 15.43953933896041027138118532153, 16.74806572634127406943706994687, 17.011648669146738728196757193742, 17.61287292722218115519294315260, 18.67714847963522583089856547834, 19.82980052536180721149772393054, 20.74597245919729930614785178861, 21.178211688467881008238256044845

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