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Properties

Label 1-319-319.214-r1-0-0
Degree $1$
Conductor $319$
Sign $0.816 - 0.577i$
Analytic cond. $34.2813$
Root an. cond. $34.2813$
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  + (−0.266 + 0.963i)2-s + (−0.512 − 0.858i)3-s + (−0.858 − 0.512i)4-s + (0.0448 + 0.998i)5-s + (0.963 − 0.266i)6-s + (−0.995 − 0.0896i)7-s + (0.722 − 0.691i)8-s + (−0.473 + 0.880i)9-s + (−0.974 − 0.222i)10-s + i·12-s + (−0.983 + 0.178i)13-s + (0.351 − 0.936i)14-s + (0.834 − 0.550i)15-s + (0.473 + 0.880i)16-s + (−0.587 + 0.809i)17-s + (−0.722 − 0.691i)18-s + ⋯
L(s)  = 1  + (−0.266 + 0.963i)2-s + (−0.512 − 0.858i)3-s + (−0.858 − 0.512i)4-s + (0.0448 + 0.998i)5-s + (0.963 − 0.266i)6-s + (−0.995 − 0.0896i)7-s + (0.722 − 0.691i)8-s + (−0.473 + 0.880i)9-s + (−0.974 − 0.222i)10-s + i·12-s + (−0.983 + 0.178i)13-s + (0.351 − 0.936i)14-s + (0.834 − 0.550i)15-s + (0.473 + 0.880i)16-s + (−0.587 + 0.809i)17-s + (−0.722 − 0.691i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 319 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 319 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(319\)    =    \(11 \cdot 29\)
Sign: $0.816 - 0.577i$
Analytic conductor: \(34.2813\)
Root analytic conductor: \(34.2813\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{319} (214, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 319,\ (1:\ ),\ 0.816 - 0.577i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2992122885 - 0.09516156968i\)
\(L(\frac12)\) \(\approx\) \(0.2992122885 - 0.09516156968i\)
\(L(1)\) \(\approx\) \(0.4874166386 + 0.1936049225i\)
\(L(1)\) \(\approx\) \(0.4874166386 + 0.1936049225i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.266 + 0.963i)T \)
3 \( 1 + (-0.512 - 0.858i)T \)
5 \( 1 + (0.0448 + 0.998i)T \)
7 \( 1 + (-0.995 - 0.0896i)T \)
13 \( 1 + (-0.983 + 0.178i)T \)
17 \( 1 + (-0.587 + 0.809i)T \)
19 \( 1 + (0.0896 + 0.995i)T \)
23 \( 1 + (0.623 - 0.781i)T \)
31 \( 1 + (-0.266 + 0.963i)T \)
37 \( 1 + (-0.990 + 0.134i)T \)
41 \( 1 + (-0.951 + 0.309i)T \)
43 \( 1 + (-0.781 - 0.623i)T \)
47 \( 1 + (0.990 + 0.134i)T \)
53 \( 1 + (-0.963 - 0.266i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + (0.919 + 0.393i)T \)
67 \( 1 + (0.900 - 0.433i)T \)
71 \( 1 + (-0.473 - 0.880i)T \)
73 \( 1 + (0.834 - 0.550i)T \)
79 \( 1 + (-0.880 - 0.473i)T \)
83 \( 1 + (0.753 - 0.657i)T \)
89 \( 1 + (0.781 - 0.623i)T \)
97 \( 1 + (0.919 - 0.393i)T \)
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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

−25.20034382961276449879684984356, −23.90202360640540789941019273474, −22.88277738979691354008273922035, −22.11298708482500029726826702285, −21.52608879895284846948166654906, −20.39562472776424186860677212684, −19.97055433618041404324613543244, −18.94562897085992518997180885814, −17.570470940611876268602104033577, −17.07300609183563677603080331430, −16.13712492872911329093557998765, −15.26785652362601728945661848519, −13.68691916725377180935228528042, −12.8498777364702965599123881592, −11.99250196089131677495040283294, −11.20066106039486513812017801352, −9.93110595187241960821141947086, −9.44318101400340082101828585209, −8.69854830439717392651108625837, −7.092374335829949430318060447252, −5.431972709785842183723364822231, −4.75344241686367640309886373827, −3.658128607846882168032357749830, −2.52031303420189397801530273669, −0.692440018659118622095461733719, 0.1718168501562261982818911821, 1.93743976486308394248172741160, 3.45805661286494633470776029747, 5.05233799673687875613341334018, 6.243471302840111568245196727324, 6.73703568780181955412710047122, 7.50693570944019978913990098529, 8.669754017090693682209412886386, 10.03705206870942214004350170165, 10.66129118618647285243658897850, 12.16341786099888082281945082095, 13.03338367919743800557759229095, 14.01112807786261078280151707785, 14.796349506386369380591257818690, 15.88481291732382815565631006049, 16.9015158941251096565277756617, 17.487290346230932438376666273604, 18.62869192656591419030200473077, 19.007531670207400436928605439576, 19.85090584900127042771285357511, 21.93694422857572144932090866965, 22.4046160232962751400636022295, 23.18004167316818433532269054952, 23.91977088802996322270900765004, 24.99692200295560265902810221758

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