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Properties

Label 1-1441-1441.1009-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.648 - 0.761i$
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  + (−0.998 + 0.0483i)2-s + (0.0241 + 0.999i)3-s + (0.995 − 0.0965i)4-s + (0.958 + 0.285i)5-s + (−0.0724 − 0.997i)6-s + (−0.926 − 0.377i)7-s + (−0.989 + 0.144i)8-s + (−0.998 + 0.0483i)9-s + (−0.970 − 0.239i)10-s + (0.120 + 0.992i)12-s + (0.0724 − 0.997i)13-s + (0.943 + 0.331i)14-s + (−0.262 + 0.964i)15-s + (0.981 − 0.192i)16-s + (0.981 − 0.192i)17-s + (0.995 − 0.0965i)18-s + ⋯
L(s)  = 1  + (−0.998 + 0.0483i)2-s + (0.0241 + 0.999i)3-s + (0.995 − 0.0965i)4-s + (0.958 + 0.285i)5-s + (−0.0724 − 0.997i)6-s + (−0.926 − 0.377i)7-s + (−0.989 + 0.144i)8-s + (−0.998 + 0.0483i)9-s + (−0.970 − 0.239i)10-s + (0.120 + 0.992i)12-s + (0.0724 − 0.997i)13-s + (0.943 + 0.331i)14-s + (−0.262 + 0.964i)15-s + (0.981 − 0.192i)16-s + (0.981 − 0.192i)17-s + (0.995 − 0.0965i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.648 - 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.648 - 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.648 - 0.761i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.648 - 0.761i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5795967749 - 0.2678019689i\)
\(L(\frac12)\) \(\approx\) \(0.5795967749 - 0.2678019689i\)
\(L(1)\) \(\approx\) \(0.6509741114 + 0.1108810769i\)
\(L(1)\) \(\approx\) \(0.6509741114 + 0.1108810769i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.998 + 0.0483i)T \)
3 \( 1 + (0.0241 + 0.999i)T \)
5 \( 1 + (0.958 + 0.285i)T \)
7 \( 1 + (-0.926 - 0.377i)T \)
13 \( 1 + (0.0724 - 0.997i)T \)
17 \( 1 + (0.981 - 0.192i)T \)
19 \( 1 + (-0.681 - 0.732i)T \)
23 \( 1 + (-0.885 + 0.464i)T \)
29 \( 1 + (0.836 - 0.548i)T \)
31 \( 1 + (-0.715 + 0.698i)T \)
37 \( 1 + (-0.995 + 0.0965i)T \)
41 \( 1 + (0.681 + 0.732i)T \)
43 \( 1 + (-0.568 + 0.822i)T \)
47 \( 1 + (-0.836 - 0.548i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (-0.681 + 0.732i)T \)
61 \( 1 + (-0.309 - 0.951i)T \)
67 \( 1 + (-0.568 - 0.822i)T \)
71 \( 1 + (0.168 - 0.985i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (-0.607 - 0.794i)T \)
83 \( 1 + (0.399 - 0.916i)T \)
89 \( 1 + T \)
97 \( 1 + (0.0724 - 0.997i)T \)
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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

−20.65880098032844981487054944613, −19.85744389772962370678611441082, −19.01404937753128844613313130250, −18.689798396979821096723000943413, −17.9700353543917575197804890984, −17.06392486175452925021308509010, −16.63939124029098949930520221919, −15.91305221091262283234721651074, −14.58791167046106783103167613165, −14.050093906331068794717395062853, −12.96023899017411800574429210699, −12.356895872617139511142981902540, −11.83595913508372626211041071473, −10.57197448268747430050028562230, −9.95642777780389321690477981754, −9.03431523493982759012357102127, −8.598475817838816667685829683644, −7.5724148558658175580203104505, −6.65908471724695963647702212539, −6.15598217289949835865215209263, −5.49912936188202771140339281702, −3.70520247965482176408928580993, −2.58717815709371031002997290263, −1.95792494394040423466921114234, −1.12833468881035469594095290746, 0.3432046665913611005767959586, 1.78424367056935742567159321407, 3.03240780720494734434006646932, 3.28811251006018604013335397707, 4.86055760543872690647666091434, 5.86927146411373273514142997419, 6.33787419739550291793520969167, 7.41327621752101071303421543829, 8.38799522289929577620750081010, 9.22807840729993935395372031409, 9.93600935304242651042461772675, 10.26883676449985880159006340976, 10.94045668661575320584786856412, 12.00494002654712557088090132073, 12.9995252247972626604253817846, 13.94424904931142438886540689249, 14.795406381623457059993956957064, 15.5141605323881991537305188547, 16.29029825148047993064020904865, 16.797771204300760825440038623372, 17.653394386772028730137362559168, 18.078902539546793792625472853805, 19.309922149585271378041140565077, 19.77649543821577002261429265108, 20.564658140241577479731382763026

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