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Properties

Label 1-1441-1441.1051-r1-0-0
Degree $1$
Conductor $1441$
Sign $-0.542 + 0.839i$
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  + (−0.958 + 0.285i)2-s + (0.885 + 0.464i)3-s + (0.836 − 0.548i)4-s + (0.885 + 0.464i)5-s + (−0.981 − 0.192i)6-s + (−0.981 + 0.192i)7-s + (−0.644 + 0.764i)8-s + (0.568 + 0.822i)9-s + (−0.981 − 0.192i)10-s + (0.995 − 0.0965i)12-s + (0.681 − 0.732i)13-s + (0.885 − 0.464i)14-s + (0.568 + 0.822i)15-s + (0.399 − 0.916i)16-s + (0.861 − 0.506i)17-s + (−0.779 − 0.626i)18-s + ⋯
L(s)  = 1  + (−0.958 + 0.285i)2-s + (0.885 + 0.464i)3-s + (0.836 − 0.548i)4-s + (0.885 + 0.464i)5-s + (−0.981 − 0.192i)6-s + (−0.981 + 0.192i)7-s + (−0.644 + 0.764i)8-s + (0.568 + 0.822i)9-s + (−0.981 − 0.192i)10-s + (0.995 − 0.0965i)12-s + (0.681 − 0.732i)13-s + (0.885 − 0.464i)14-s + (0.568 + 0.822i)15-s + (0.399 − 0.916i)16-s + (0.861 − 0.506i)17-s + (−0.779 − 0.626i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.542 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.542 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.542 + 0.839i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1051, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ -0.542 + 0.839i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9748787982 + 1.790518247i\)
\(L(\frac12)\) \(\approx\) \(0.9748787982 + 1.790518247i\)
\(L(1)\) \(\approx\) \(0.9731665187 + 0.4594679874i\)
\(L(1)\) \(\approx\) \(0.9731665187 + 0.4594679874i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.958 + 0.285i)T \)
3 \( 1 + (0.885 + 0.464i)T \)
5 \( 1 + (0.885 + 0.464i)T \)
7 \( 1 + (-0.981 + 0.192i)T \)
13 \( 1 + (0.681 - 0.732i)T \)
17 \( 1 + (0.861 - 0.506i)T \)
19 \( 1 + (0.861 + 0.506i)T \)
23 \( 1 + (-0.0724 - 0.997i)T \)
29 \( 1 + (-0.958 - 0.285i)T \)
31 \( 1 + (-0.970 + 0.239i)T \)
37 \( 1 + (-0.998 - 0.0483i)T \)
41 \( 1 + (-0.995 + 0.0965i)T \)
43 \( 1 + (0.443 + 0.896i)T \)
47 \( 1 + (-0.607 + 0.794i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (0.215 + 0.976i)T \)
61 \( 1 + (-0.309 + 0.951i)T \)
67 \( 1 + (-0.443 + 0.896i)T \)
71 \( 1 + (0.926 + 0.377i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (0.989 - 0.144i)T \)
83 \( 1 + (0.354 + 0.935i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (0.120 - 0.992i)T \)
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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

−20.162758437509960907727760260835, −19.54534512958601747893533741177, −18.68562641255350202407270229621, −18.34252283729256955516343727197, −17.29930690279872471459748394629, −16.67711061942172714635934021163, −15.92205312626364293704021316798, −15.12754787473095783650437941830, −13.89483837876550678272951894646, −13.452123638144414635215866683186, −12.62211681012548157563380726600, −11.99883577803094330440825682949, −10.807702519519899262107658488309, −9.882084910932270874811473165433, −9.32319370760569179689223589250, −8.916721803294979259403372654799, −7.89353365451923902093956039811, −7.07033052807515886296135216600, −6.40634866987805671204823681810, −5.45954508363121038809022926763, −3.65165750108759337126869637244, −3.33362974406884002790742562519, −2.021794996544537868371327374502, −1.5291147489876396954697550910, −0.4786566798356858895323634348, 1.05090705924772540536266870883, 2.0952691251856216049528009918, 2.99501013933787095704834650002, 3.51027190554686322914629020415, 5.28397310589463900339295395631, 5.87015328560598768555521982212, 6.86201136692944303262012284436, 7.59019431943330667856282677659, 8.53141831531169569777392465752, 9.25771624530595710161849565552, 9.93186106444971958394976626289, 10.29477402447218221460670791219, 11.20723430192363378734278282333, 12.49115372665497550088840085032, 13.347703205850041507680106643539, 14.19722179196857124873355833566, 14.79539064720256250751676884320, 15.62735489071203390341509111482, 16.33467815398224015537075348944, 16.8126254982907515829149287337, 18.15704281066296718628142737064, 18.41093509839139746307240151440, 19.19226969238078896762522474624, 19.97800330127275748744306453147, 20.78419250430139587744639731431

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