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Properties

Label 1-389-389.7-r0-0-0
Degree $1$
Conductor $389$
Sign $0.971 - 0.236i$
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  + (0.563 − 0.825i)2-s + (0.834 + 0.550i)3-s + (−0.363 − 0.931i)4-s + (0.997 + 0.0647i)5-s + (0.925 − 0.378i)6-s + (−0.177 + 0.984i)7-s + (−0.974 − 0.224i)8-s + (0.393 + 0.919i)9-s + (0.616 − 0.787i)10-s + (−0.816 + 0.577i)11-s + (0.208 − 0.977i)12-s + (0.948 + 0.318i)13-s + (0.712 + 0.701i)14-s + (0.797 + 0.603i)15-s + (−0.735 + 0.677i)16-s + (0.616 − 0.787i)17-s + ⋯
L(s)  = 1  + (0.563 − 0.825i)2-s + (0.834 + 0.550i)3-s + (−0.363 − 0.931i)4-s + (0.997 + 0.0647i)5-s + (0.925 − 0.378i)6-s + (−0.177 + 0.984i)7-s + (−0.974 − 0.224i)8-s + (0.393 + 0.919i)9-s + (0.616 − 0.787i)10-s + (−0.816 + 0.577i)11-s + (0.208 − 0.977i)12-s + (0.948 + 0.318i)13-s + (0.712 + 0.701i)14-s + (0.797 + 0.603i)15-s + (−0.735 + 0.677i)16-s + (0.616 − 0.787i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(389\)
Sign: $0.971 - 0.236i$
Analytic conductor: \(1.80650\)
Root analytic conductor: \(1.80650\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{389} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 389,\ (0:\ ),\ 0.971 - 0.236i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.474076067 - 0.2964200296i\)
\(L(\frac12)\) \(\approx\) \(2.474076067 - 0.2964200296i\)
\(L(1)\) \(\approx\) \(1.882203377 - 0.3046030205i\)
\(L(1)\) \(\approx\) \(1.882203377 - 0.3046030205i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad389 \( 1 \)
good2 \( 1 + (0.563 - 0.825i)T \)
3 \( 1 + (0.834 + 0.550i)T \)
5 \( 1 + (0.997 + 0.0647i)T \)
7 \( 1 + (-0.177 + 0.984i)T \)
11 \( 1 + (-0.816 + 0.577i)T \)
13 \( 1 + (0.948 + 0.318i)T \)
17 \( 1 + (0.616 - 0.787i)T \)
19 \( 1 + (-0.995 + 0.0970i)T \)
23 \( 1 + (0.997 - 0.0647i)T \)
29 \( 1 + (-0.113 - 0.993i)T \)
31 \( 1 + (0.898 + 0.438i)T \)
37 \( 1 + (-0.689 - 0.724i)T \)
41 \( 1 + (-0.995 - 0.0970i)T \)
43 \( 1 + (-0.536 - 0.843i)T \)
47 \( 1 + (-0.995 - 0.0970i)T \)
53 \( 1 + (0.0161 - 0.999i)T \)
59 \( 1 + (-0.590 + 0.807i)T \)
61 \( 1 + (0.948 + 0.318i)T \)
67 \( 1 + (0.756 + 0.653i)T \)
71 \( 1 + (-0.363 - 0.931i)T \)
73 \( 1 + (-0.735 - 0.677i)T \)
79 \( 1 + (0.898 - 0.438i)T \)
83 \( 1 + (0.756 - 0.653i)T \)
89 \( 1 + (0.333 - 0.942i)T \)
97 \( 1 + (-0.363 + 0.931i)T \)
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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

−24.582948745541726044360873859, −23.569038200936025253253523177752, −23.2670143367601907913189676254, −21.80356190214607492533494930969, −20.99240697231056270345939415080, −20.53713996806365778339227062076, −19.09823923447013313954327106325, −18.27887194073385818630517462295, −17.32582882441503985002187242816, −16.63997062633380233202269633106, −15.47125690113894000134230478830, −14.57190328206115853417138988592, −13.71764053085365424871232065373, −13.19845995725350980388712229487, −12.67085694056110409539966575271, −10.899538678285037074100753255783, −9.840014870591271046980620557974, −8.57930432799978766154731821794, −8.07247796212401787828970159341, −6.80541249471784240617931706975, −6.22522501087192559963139645145, −5.01614632513458776202222654582, −3.645894459626246875758505200067, −2.8756951860240417104340066443, −1.307375457614164605240683930479, 1.78481515138549502307318285697, 2.51177228334318218866131728579, 3.38606835259064849339253233699, 4.78445326775946575160666862308, 5.46869068792710831705231476353, 6.64648213918654316589856487660, 8.444657997732932315598030159461, 9.18969058973389134965739497913, 10.009391384256462391911106735830, 10.69647588480738441161831398685, 11.967175748611235278774305778407, 13.11554672245937509983522526601, 13.53744760599870127327001470673, 14.62636852829257220274677450245, 15.256385912216254828330147967610, 16.19528505927415054571350828013, 17.71560389506533932249825146328, 18.722930980730313539578344727592, 19.11146485833670310576075064795, 20.55229968479370133575073577968, 21.069028485520696143387735579844, 21.4250268105568605099060219464, 22.50092733110946783981387772906, 23.253523912675709767478627079242, 24.64552681448537326307662823588

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