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Properties

Label 1-389-389.171-r0-0-0
Degree $1$
Conductor $389$
Sign $0.259 + 0.965i$
Analytic cond. $1.80650$
Root an. cond. $1.80650$
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  + (0.898 + 0.438i)2-s + (−0.937 − 0.348i)3-s + (0.616 + 0.787i)4-s + (0.925 + 0.378i)5-s + (−0.689 − 0.724i)6-s + (−0.481 − 0.876i)7-s + (0.208 + 0.977i)8-s + (0.756 + 0.653i)9-s + (0.665 + 0.746i)10-s + (−0.852 + 0.523i)11-s + (−0.302 − 0.953i)12-s + (−0.363 + 0.931i)13-s + (−0.0485 − 0.998i)14-s + (−0.735 − 0.677i)15-s + (−0.240 + 0.970i)16-s + (0.665 + 0.746i)17-s + ⋯
L(s)  = 1  + (0.898 + 0.438i)2-s + (−0.937 − 0.348i)3-s + (0.616 + 0.787i)4-s + (0.925 + 0.378i)5-s + (−0.689 − 0.724i)6-s + (−0.481 − 0.876i)7-s + (0.208 + 0.977i)8-s + (0.756 + 0.653i)9-s + (0.665 + 0.746i)10-s + (−0.852 + 0.523i)11-s + (−0.302 − 0.953i)12-s + (−0.363 + 0.931i)13-s + (−0.0485 − 0.998i)14-s + (−0.735 − 0.677i)15-s + (−0.240 + 0.970i)16-s + (0.665 + 0.746i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.259 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.259 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(389\)
Sign: $0.259 + 0.965i$
Analytic conductor: \(1.80650\)
Root analytic conductor: \(1.80650\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{389} (171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 389,\ (0:\ ),\ 0.259 + 0.965i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.406434705 + 1.078951528i\)
\(L(\frac12)\) \(\approx\) \(1.406434705 + 1.078951528i\)
\(L(1)\) \(\approx\) \(1.357194477 + 0.5087862087i\)
\(L(1)\) \(\approx\) \(1.357194477 + 0.5087862087i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad389 \( 1 \)
good2 \( 1 + (0.898 + 0.438i)T \)
3 \( 1 + (-0.937 - 0.348i)T \)
5 \( 1 + (0.925 + 0.378i)T \)
7 \( 1 + (-0.481 - 0.876i)T \)
11 \( 1 + (-0.852 + 0.523i)T \)
13 \( 1 + (-0.363 + 0.931i)T \)
17 \( 1 + (0.665 + 0.746i)T \)
19 \( 1 + (0.834 - 0.550i)T \)
23 \( 1 + (0.925 - 0.378i)T \)
29 \( 1 + (-0.777 + 0.628i)T \)
31 \( 1 + (-0.912 + 0.408i)T \)
37 \( 1 + (0.145 - 0.989i)T \)
41 \( 1 + (0.834 + 0.550i)T \)
43 \( 1 + (0.966 - 0.256i)T \)
47 \( 1 + (0.834 + 0.550i)T \)
53 \( 1 + (-0.995 - 0.0970i)T \)
59 \( 1 + (0.797 + 0.603i)T \)
61 \( 1 + (-0.363 + 0.931i)T \)
67 \( 1 + (-0.423 - 0.905i)T \)
71 \( 1 + (0.616 + 0.787i)T \)
73 \( 1 + (-0.240 - 0.970i)T \)
79 \( 1 + (-0.912 - 0.408i)T \)
83 \( 1 + (-0.423 + 0.905i)T \)
89 \( 1 + (0.452 - 0.891i)T \)
97 \( 1 + (0.616 - 0.787i)T \)
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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

−24.23931768774943278098719063357, −23.13533684176239104386321191451, −22.395284261197498196170169617583, −21.85002574268101646419502477255, −20.95055690902149635055159443948, −20.48297379574197722564931864665, −18.856700750960316468414083085987, −18.3388126717458221196769018567, −17.13051390617821647368668525927, −16.134390120965024309868521199665, −15.55320271611636942229139831761, −14.473826861560029354759355703091, −13.22567098001501477559601535731, −12.72828572700784309220811196458, −11.848062576979613641759335267251, −10.86694461592250878194883787954, −9.91856026931765465512281999324, −9.34848687368424213109391620815, −7.471570308791498884983750747945, −6.07430526091075911156997135601, −5.48673508600827566960266343091, −5.07793970093923981020702886751, −3.42348051955618861838410129731, −2.45941526677532000406350171740, −0.925274172553523713852519231678, 1.60964532530652900938849459646, 2.85036732478453841671589426201, 4.26141792460668028413494094881, 5.26759361011238563394930093473, 6.03195229211545891404958798933, 7.16157480368514004988782801409, 7.358482452935327537156194017095, 9.3695243185654192330916407931, 10.54675469170884600335351678320, 11.11501600802711937847537602596, 12.54247885263504005758556130354, 12.962586404395277014276629031149, 13.89167019595223466133624603300, 14.70416355822332217981956531377, 16.02219438973973125423322376721, 16.70467166122093177271687950719, 17.394651728500901437207534784450, 18.23341316556598338337913327633, 19.3917864145560555963501441062, 20.69118897252885781836533213858, 21.44638262646743745081108345979, 22.256278519616886892051397576243, 22.93403542456918497091171087964, 23.71521466229405384702140968345, 24.32169022890051402346331348097

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