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Properties

Label 1-389-389.78-r0-0-0
Degree $1$
Conductor $389$
Sign $-0.296 - 0.954i$
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  + (0.208 − 0.977i)2-s + (−0.481 + 0.876i)3-s + (−0.912 − 0.408i)4-s + (0.393 − 0.919i)5-s + (0.756 + 0.653i)6-s + (0.997 − 0.0647i)7-s + (−0.590 + 0.807i)8-s + (−0.536 − 0.843i)9-s + (−0.816 − 0.577i)10-s + (0.0808 − 0.996i)11-s + (0.797 − 0.603i)12-s + (0.898 + 0.438i)13-s + (0.145 − 0.989i)14-s + (0.616 + 0.787i)15-s + (0.665 + 0.746i)16-s + (−0.816 − 0.577i)17-s + ⋯
L(s)  = 1  + (0.208 − 0.977i)2-s + (−0.481 + 0.876i)3-s + (−0.912 − 0.408i)4-s + (0.393 − 0.919i)5-s + (0.756 + 0.653i)6-s + (0.997 − 0.0647i)7-s + (−0.590 + 0.807i)8-s + (−0.536 − 0.843i)9-s + (−0.816 − 0.577i)10-s + (0.0808 − 0.996i)11-s + (0.797 − 0.603i)12-s + (0.898 + 0.438i)13-s + (0.145 − 0.989i)14-s + (0.616 + 0.787i)15-s + (0.665 + 0.746i)16-s + (−0.816 − 0.577i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.296 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.296 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(389\)
Sign: $-0.296 - 0.954i$
Analytic conductor: \(1.80650\)
Root analytic conductor: \(1.80650\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{389} (78, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 389,\ (0:\ ),\ -0.296 - 0.954i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7299914792 - 0.9911844727i\)
\(L(\frac12)\) \(\approx\) \(0.7299914792 - 0.9911844727i\)
\(L(1)\) \(\approx\) \(0.9099633470 - 0.5411643695i\)
\(L(1)\) \(\approx\) \(0.9099633470 - 0.5411643695i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad389 \( 1 \)
good2 \( 1 + (0.208 - 0.977i)T \)
3 \( 1 + (-0.481 + 0.876i)T \)
5 \( 1 + (0.393 - 0.919i)T \)
7 \( 1 + (0.997 - 0.0647i)T \)
11 \( 1 + (0.0808 - 0.996i)T \)
13 \( 1 + (0.898 + 0.438i)T \)
17 \( 1 + (-0.816 - 0.577i)T \)
19 \( 1 + (-0.177 + 0.984i)T \)
23 \( 1 + (0.393 + 0.919i)T \)
29 \( 1 + (0.452 - 0.891i)T \)
31 \( 1 + (-0.302 - 0.953i)T \)
37 \( 1 + (-0.423 - 0.905i)T \)
41 \( 1 + (-0.177 - 0.984i)T \)
43 \( 1 + (0.712 + 0.701i)T \)
47 \( 1 + (-0.177 - 0.984i)T \)
53 \( 1 + (-0.957 + 0.287i)T \)
59 \( 1 + (-0.363 - 0.931i)T \)
61 \( 1 + (0.898 + 0.438i)T \)
67 \( 1 + (0.966 - 0.256i)T \)
71 \( 1 + (-0.912 - 0.408i)T \)
73 \( 1 + (0.665 - 0.746i)T \)
79 \( 1 + (-0.302 + 0.953i)T \)
83 \( 1 + (0.966 + 0.256i)T \)
89 \( 1 + (-0.986 - 0.161i)T \)
97 \( 1 + (-0.912 + 0.408i)T \)
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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

−24.725240758751749269985929415303, −23.80021169102998827060055711956, −23.22435659882790474650031530303, −22.33999232948860206247813935867, −21.71111394351525674257904645493, −20.41643997956817258324505047564, −19.081584555703148576628441300617, −18.1022816250794174939175951348, −17.781683971356679331820435176056, −17.169475476378900715991843124813, −15.77262460767791414208520167487, −14.880277051224255241611162642357, −14.17562146060578645064093219562, −13.26578290583171498071063510283, −12.48290452752695745808969669150, −11.2175423925702086215295926801, −10.466359668790550979939674114560, −8.865556338178507337696341043930, −8.034355664097840038147162146550, −6.93557823123463849135729381047, −6.54359151578149300765157431170, −5.375939224787848818820465802030, −4.48492044696004417435634897333, −2.808669202601365282109405084388, −1.478969556852086126809905637438, 0.81054591368650076562961502367, 2.00588245617891126775674282867, 3.65863043341260019356479097608, 4.38279154520789647275025033391, 5.35557736609979108853428440345, 6.01229105567422513185947214883, 8.26233429424770719506498884454, 8.95246468806855467863253965403, 9.754518125824786970327950617315, 10.99944199806455193718835792775, 11.3491236845663864895426346702, 12.325499614702237636875492230187, 13.58859807261349567979936018364, 14.10830663410072721181585758230, 15.375115395650164896025132807627, 16.38340693134597186570750599146, 17.295483128263586611013711484447, 17.975513183074332364772812017058, 19.08054233152780028683648657250, 20.29977004966176117218611944071, 20.97652336108663505711388893260, 21.28045432198839000678451701103, 22.19127655943618010752675492778, 23.26100311584809810877156387035, 23.93149587359715465845720844970

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