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Properties

Label 1-389-389.6-r0-0-0
Degree $1$
Conductor $389$
Sign $-0.408 - 0.912i$
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  + (−0.302 − 0.953i)2-s + (0.991 − 0.129i)3-s + (−0.816 + 0.577i)4-s + (0.756 − 0.653i)5-s + (−0.423 − 0.905i)6-s + (0.925 − 0.378i)7-s + (0.797 + 0.603i)8-s + (0.966 − 0.256i)9-s + (−0.852 − 0.523i)10-s + (−0.884 − 0.466i)11-s + (−0.735 + 0.677i)12-s + (−0.912 + 0.408i)13-s + (−0.641 − 0.767i)14-s + (0.665 − 0.746i)15-s + (0.333 − 0.942i)16-s + (−0.852 − 0.523i)17-s + ⋯
L(s)  = 1  + (−0.302 − 0.953i)2-s + (0.991 − 0.129i)3-s + (−0.816 + 0.577i)4-s + (0.756 − 0.653i)5-s + (−0.423 − 0.905i)6-s + (0.925 − 0.378i)7-s + (0.797 + 0.603i)8-s + (0.966 − 0.256i)9-s + (−0.852 − 0.523i)10-s + (−0.884 − 0.466i)11-s + (−0.735 + 0.677i)12-s + (−0.912 + 0.408i)13-s + (−0.641 − 0.767i)14-s + (0.665 − 0.746i)15-s + (0.333 − 0.942i)16-s + (−0.852 − 0.523i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.408 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.408 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(389\)
Sign: $-0.408 - 0.912i$
Analytic conductor: \(1.80650\)
Root analytic conductor: \(1.80650\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{389} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 389,\ (0:\ ),\ -0.408 - 0.912i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9203120102 - 1.419739552i\)
\(L(\frac12)\) \(\approx\) \(0.9203120102 - 1.419739552i\)
\(L(1)\) \(\approx\) \(1.069727085 - 0.8181577921i\)
\(L(1)\) \(\approx\) \(1.069727085 - 0.8181577921i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad389 \( 1 \)
good2 \( 1 + (-0.302 - 0.953i)T \)
3 \( 1 + (0.991 - 0.129i)T \)
5 \( 1 + (0.756 - 0.653i)T \)
7 \( 1 + (0.925 - 0.378i)T \)
11 \( 1 + (-0.884 - 0.466i)T \)
13 \( 1 + (-0.912 + 0.408i)T \)
17 \( 1 + (-0.852 - 0.523i)T \)
19 \( 1 + (-0.481 - 0.876i)T \)
23 \( 1 + (0.756 + 0.653i)T \)
29 \( 1 + (0.948 - 0.318i)T \)
31 \( 1 + (0.271 + 0.962i)T \)
37 \( 1 + (0.868 + 0.495i)T \)
41 \( 1 + (-0.481 + 0.876i)T \)
43 \( 1 + (-0.0485 - 0.998i)T \)
47 \( 1 + (-0.481 + 0.876i)T \)
53 \( 1 + (-0.177 - 0.984i)T \)
59 \( 1 + (0.616 + 0.787i)T \)
61 \( 1 + (-0.912 + 0.408i)T \)
67 \( 1 + (0.0161 - 0.999i)T \)
71 \( 1 + (-0.816 + 0.577i)T \)
73 \( 1 + (0.333 + 0.942i)T \)
79 \( 1 + (0.271 - 0.962i)T \)
83 \( 1 + (0.0161 + 0.999i)T \)
89 \( 1 + (0.563 + 0.825i)T \)
97 \( 1 + (-0.816 - 0.577i)T \)
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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

−24.94410487144249826579493288927, −24.26948824761211355847501909055, −23.17258677624612790059565585036, −22.0705568450850402332511157803, −21.36257998376807552833690467648, −20.41407505356911152666804386483, −19.24889058617940549327334909295, −18.4343736602677616632222955144, −17.81999257909067004356124699888, −16.94081082383630587206991685477, −15.57497179475293808920698183934, −14.85165308115889650447672518231, −14.5348946331401561962818876028, −13.45604353826948939728993052017, −12.65915478899542425800122503673, −10.698647739522201503261361606282, −10.12296776576668012486166281912, −9.12380542411522624683394088591, −8.179933556395936978323135510077, −7.502143237947630723241119798827, −6.43265010107297897110916383935, −5.214970925965394989994282095248, −4.37386293764503599107395665060, −2.66462017325344469843397525065, −1.80500147160390484863546115201, 1.07387614079173061365885780501, 2.17133530482376223847841959661, 2.88573735268850462791877535641, 4.53023200329223366021706199113, 4.951972370946896806723218570744, 6.99918310159309229675054571026, 8.11052492388599802093361346890, 8.77280876964694707785154600741, 9.614957449884338565479978227339, 10.50072216545622563712054398540, 11.55639907198410713741634159624, 12.725729828576009835853958415836, 13.516100220902881728768559254040, 13.94931508471268654799755769559, 15.13881465953370100788057644621, 16.46807852011009275013822283495, 17.574759656028790494743808483901, 18.02519150699863785055706545279, 19.21272139412874367873725189549, 19.921722495624699254402385249519, 20.70935784799650768828133529051, 21.381609793184986923442557815017, 21.81566208617820170895873063432, 23.52052035592488031105789888607, 24.238385540604160392764791127684

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