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Properties

Label 1-1441-1441.580-r0-0-0
Degree $1$
Conductor $1441$
Sign $-0.963 - 0.269i$
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  + (0.926 + 0.377i)2-s + (0.120 + 0.992i)3-s + (0.715 + 0.698i)4-s + (0.120 + 0.992i)5-s + (−0.262 + 0.964i)6-s + (0.262 + 0.964i)7-s + (0.399 + 0.916i)8-s + (−0.970 + 0.239i)9-s + (−0.262 + 0.964i)10-s + (−0.607 + 0.794i)12-s + (0.998 + 0.0483i)13-s + (−0.120 + 0.992i)14-s + (−0.970 + 0.239i)15-s + (0.0241 + 0.999i)16-s + (−0.943 + 0.331i)17-s + (−0.989 − 0.144i)18-s + ⋯
L(s)  = 1  + (0.926 + 0.377i)2-s + (0.120 + 0.992i)3-s + (0.715 + 0.698i)4-s + (0.120 + 0.992i)5-s + (−0.262 + 0.964i)6-s + (0.262 + 0.964i)7-s + (0.399 + 0.916i)8-s + (−0.970 + 0.239i)9-s + (−0.262 + 0.964i)10-s + (−0.607 + 0.794i)12-s + (0.998 + 0.0483i)13-s + (−0.120 + 0.992i)14-s + (−0.970 + 0.239i)15-s + (0.0241 + 0.999i)16-s + (−0.943 + 0.331i)17-s + (−0.989 − 0.144i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.963 - 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.963 - 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.963 - 0.269i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (580, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ -0.963 - 0.269i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3870093140 + 2.822496277i\)
\(L(\frac12)\) \(\approx\) \(-0.3870093140 + 2.822496277i\)
\(L(1)\) \(\approx\) \(1.074442649 + 1.553643104i\)
\(L(1)\) \(\approx\) \(1.074442649 + 1.553643104i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.926 + 0.377i)T \)
3 \( 1 + (0.120 + 0.992i)T \)
5 \( 1 + (0.120 + 0.992i)T \)
7 \( 1 + (0.262 + 0.964i)T \)
13 \( 1 + (0.998 + 0.0483i)T \)
17 \( 1 + (-0.943 + 0.331i)T \)
19 \( 1 + (-0.943 - 0.331i)T \)
23 \( 1 + (-0.995 - 0.0965i)T \)
29 \( 1 + (0.926 - 0.377i)T \)
31 \( 1 + (0.748 - 0.663i)T \)
37 \( 1 + (0.443 - 0.896i)T \)
41 \( 1 + (0.607 - 0.794i)T \)
43 \( 1 + (0.906 - 0.421i)T \)
47 \( 1 + (-0.644 + 0.764i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.958 + 0.285i)T \)
61 \( 1 + (0.809 + 0.587i)T \)
67 \( 1 + (0.906 + 0.421i)T \)
71 \( 1 + (0.861 + 0.506i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (0.981 + 0.192i)T \)
83 \( 1 + (0.885 + 0.464i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (0.354 - 0.935i)T \)
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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

−20.25803167785121095150550425496, −19.86147402990273076774340883400, −19.10348221833607917452822750333, −18.021890707127864866051192824683, −17.41293528018303516788814034376, −16.41470380660358478299269737497, −15.833944895118593805053943528820, −14.674858048439330058387526865156, −13.84690415836079478130271928287, −13.47134563963064980189214524532, −12.81846300796177414067926891186, −12.10513446025991521752672891399, −11.304809498280598317541482992202, −10.5942376976161236323588545638, −9.52573290064160519589538761233, −8.39079191398764868930305124328, −7.86460910569677243799408198208, −6.52589753898014209120364494730, −6.333968914229120069428921366786, −5.083260765335286380481461948202, −4.359571930749898009612035345560, −3.51130148251245043321656378804, −2.298967632786947562844688892898, −1.474121548064964854930896021097, −0.741834171580464745567194203557, 2.32972815332270940221078484898, 2.48941063066198684309152788223, 3.81648526959434500559095412408, 4.21506365219617914316984135404, 5.36807954968712091273773179664, 6.12579654443644207375610655480, 6.58388745532730962244736774413, 8.02791973645772852797776828760, 8.52410488332703095464781886828, 9.55879693678073130986075602140, 10.704198451852027025997829206310, 11.11367558850630697270235219998, 11.83456243846190759220593043680, 12.88789000925562753655088085302, 13.83303001944306149983139826747, 14.42481830667627163956310309787, 15.10689857642943596378509001226, 15.71930974368673354527366077032, 16.081918905060907555114502646576, 17.47179351015642207047902991274, 17.7649862002479923603753665975, 19.03398866456517943807804338471, 19.73970592622667083322081853254, 20.91530842626207302141338182008, 21.19027356023093460072946635394

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