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Properties

Label 1-1441-1441.884-r1-0-0
Degree $1$
Conductor $1441$
Sign $0.749 + 0.662i$
Analytic cond. $154.856$
Root an. cond. $154.856$
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  + (−0.399 − 0.916i)2-s + (−0.998 + 0.0483i)3-s + (−0.681 + 0.732i)4-s + (0.836 + 0.548i)5-s + (0.443 + 0.896i)6-s + (0.885 − 0.464i)7-s + (0.943 + 0.331i)8-s + (0.995 − 0.0965i)9-s + (0.168 − 0.985i)10-s + (0.644 − 0.764i)12-s + (0.885 − 0.464i)13-s + (−0.779 − 0.626i)14-s + (−0.861 − 0.506i)15-s + (−0.0724 − 0.997i)16-s + (−0.644 − 0.764i)17-s + (−0.485 − 0.873i)18-s + ⋯
L(s)  = 1  + (−0.399 − 0.916i)2-s + (−0.998 + 0.0483i)3-s + (−0.681 + 0.732i)4-s + (0.836 + 0.548i)5-s + (0.443 + 0.896i)6-s + (0.885 − 0.464i)7-s + (0.943 + 0.331i)8-s + (0.995 − 0.0965i)9-s + (0.168 − 0.985i)10-s + (0.644 − 0.764i)12-s + (0.885 − 0.464i)13-s + (−0.779 − 0.626i)14-s + (−0.861 − 0.506i)15-s + (−0.0724 − 0.997i)16-s + (−0.644 − 0.764i)17-s + (−0.485 − 0.873i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.749 + 0.662i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (884, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ 0.749 + 0.662i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.126443280 + 0.4265952829i\)
\(L(\frac12)\) \(\approx\) \(1.126443280 + 0.4265952829i\)
\(L(1)\) \(\approx\) \(0.7737948336 - 0.1758781867i\)
\(L(1)\) \(\approx\) \(0.7737948336 - 0.1758781867i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.399 - 0.916i)T \)
3 \( 1 + (-0.998 + 0.0483i)T \)
5 \( 1 + (0.836 + 0.548i)T \)
7 \( 1 + (0.885 - 0.464i)T \)
13 \( 1 + (0.885 - 0.464i)T \)
17 \( 1 + (-0.644 - 0.764i)T \)
19 \( 1 + (-0.926 - 0.377i)T \)
23 \( 1 + (0.607 + 0.794i)T \)
29 \( 1 + (0.748 + 0.663i)T \)
31 \( 1 + (-0.0241 + 0.999i)T \)
37 \( 1 + (-0.120 + 0.992i)T \)
41 \( 1 + (-0.527 - 0.849i)T \)
43 \( 1 + (0.836 + 0.548i)T \)
47 \( 1 + (-0.995 + 0.0965i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (0.644 + 0.764i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + (-0.836 + 0.548i)T \)
71 \( 1 + (-0.958 + 0.285i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (-0.779 + 0.626i)T \)
83 \( 1 + (0.681 + 0.732i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (0.989 + 0.144i)T \)
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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

−20.709375150933811835034236642462, −19.34652667603238518918812341870, −18.56372333787719363213516982134, −17.9862636349407638440810663840, −17.35020481635313608893931560472, −16.84117515103386708977477633245, −16.14232742495997886088899820968, −15.27467784776924769605220836568, −14.57851498428493898152218633937, −13.55984204487330868594720902347, −12.98271971508323112489099302350, −12.06751443245956980555313616906, −10.93388930715668957503837166341, −10.52268521908314956661740854993, −9.45224759940046196127532781268, −8.685980273370574780231040442985, −8.092802556856013270132608652327, −6.87398864988600208960355084480, −6.029636878902544541369917474588, −5.81179268890500185358412290925, −4.587729107845626281956596359890, −4.34858322306498486298277645501, −2.03766442418792219198165493795, −1.42262157192577126581942010550, −0.3501819352797082847574801164, 0.99590652354065863474944890659, 1.56376094981001983137067201854, 2.67526821983103105901339452126, 3.73994942031918617002631342269, 4.78659261405063906410986190582, 5.32473125851401853038835462899, 6.581531584467139044076615193489, 7.19574610934769121520770366523, 8.354287887402699857144784209785, 9.1828412400881445624884343692, 10.1828505734213329626649978032, 10.72658173190972088639988666388, 11.17122756780270278848482068688, 11.91575379176504405978581523061, 13.04178491526672999112720302786, 13.457483206308632139727236917378, 14.33333150172725132549121192742, 15.42542467800363760844176985688, 16.40681505960698342815285274386, 17.298160257727303962736033738, 17.74908968344052749964157587292, 18.118925963872000595880105091305, 18.95103967132404575183123853457, 19.91151185904526225368453526718, 20.96846188084374661250246401746

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