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Properties

Label 1-1441-1441.311-r0-0-0
Degree $1$
Conductor $1441$
Sign $-0.989 + 0.146i$
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  + (−0.168 + 0.985i)2-s + (−0.0724 − 0.997i)3-s + (−0.943 − 0.331i)4-s + (0.644 + 0.764i)5-s + (0.995 + 0.0965i)6-s + (−0.748 + 0.663i)7-s + (0.485 − 0.873i)8-s + (−0.989 + 0.144i)9-s + (−0.861 + 0.506i)10-s + (−0.262 + 0.964i)12-s + (−0.748 + 0.663i)13-s + (−0.527 − 0.849i)14-s + (0.715 − 0.698i)15-s + (0.779 + 0.626i)16-s + (−0.262 − 0.964i)17-s + (0.0241 − 0.999i)18-s + ⋯
L(s)  = 1  + (−0.168 + 0.985i)2-s + (−0.0724 − 0.997i)3-s + (−0.943 − 0.331i)4-s + (0.644 + 0.764i)5-s + (0.995 + 0.0965i)6-s + (−0.748 + 0.663i)7-s + (0.485 − 0.873i)8-s + (−0.989 + 0.144i)9-s + (−0.861 + 0.506i)10-s + (−0.262 + 0.964i)12-s + (−0.748 + 0.663i)13-s + (−0.527 − 0.849i)14-s + (0.715 − 0.698i)15-s + (0.779 + 0.626i)16-s + (−0.262 − 0.964i)17-s + (0.0241 − 0.999i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.989 + 0.146i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ -0.989 + 0.146i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04460187984 + 0.6053395745i\)
\(L(\frac12)\) \(\approx\) \(0.04460187984 + 0.6053395745i\)
\(L(1)\) \(\approx\) \(0.6745515230 + 0.3294875520i\)
\(L(1)\) \(\approx\) \(0.6745515230 + 0.3294875520i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.168 + 0.985i)T \)
3 \( 1 + (-0.0724 - 0.997i)T \)
5 \( 1 + (0.644 + 0.764i)T \)
7 \( 1 + (-0.748 + 0.663i)T \)
13 \( 1 + (-0.748 + 0.663i)T \)
17 \( 1 + (-0.262 - 0.964i)T \)
19 \( 1 + (0.836 + 0.548i)T \)
23 \( 1 + (0.981 - 0.192i)T \)
29 \( 1 + (0.885 - 0.464i)T \)
31 \( 1 + (-0.681 - 0.732i)T \)
37 \( 1 + (0.568 + 0.822i)T \)
41 \( 1 + (-0.998 + 0.0483i)T \)
43 \( 1 + (0.644 + 0.764i)T \)
47 \( 1 + (-0.989 + 0.144i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (-0.262 - 0.964i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + (0.644 - 0.764i)T \)
71 \( 1 + (-0.906 + 0.421i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (-0.527 + 0.849i)T \)
83 \( 1 + (-0.943 + 0.331i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (0.215 - 0.976i)T \)
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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

−20.202702126661355214746969404, −19.95463245109337135014579968990, −19.261708214363953174419902540899, −17.81061627679509929838110675983, −17.469764996741850502483859764543, −16.694405549654992907456008328703, −16.1058983137075527443749093185, −14.996046535698378690729564421910, −14.12557083883568907570987929023, −13.31972398399687982000424265452, −12.7263748277629364114707476977, −11.9363346509258311545326398645, −10.81713270751793405665877901796, −10.33762621807451832823688023958, −9.65732054506381341744232756021, −9.05960479529967819889183506832, −8.33133561042299317717994577006, −7.0963404889606488736956051800, −5.744561876252912555545513573837, −5.05129789097657535243996045265, −4.341267647984063605537876347511, −3.3752620630880601288076206059, −2.72288427070858819856674229459, −1.40056887289741224881918019020, −0.26444679359707731865619530045, 1.276791222891087146113792057780, 2.49019343290455559341158418698, 3.17175299174739130302942557186, 4.76492185301076302173630744009, 5.64819707232588617784277544384, 6.29991802344254208868092560374, 6.95602612420401135979952000083, 7.48559456618880953514923042046, 8.56406695684688424177168512492, 9.47268945484106966697358519558, 9.86149268516769872476067208160, 11.204869655845277216491771417671, 12.04935899012780979239252957686, 12.971648630839577911818199108840, 13.59727142018650991155934799395, 14.2776707360422043880258271844, 14.90172807920175212842608141151, 15.83240902786145889808097075055, 16.74796152672906927034286545360, 17.27686465795754188212831563410, 18.36508273117086779382779922854, 18.45251532540660999730424683656, 19.18337003650841412814112664440, 20.001944874745211875579443879032, 21.34817473751454466754641692484

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