Properties

Label 1-389-389.65-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: $\chi_{389} (65, \cdot )$
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

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

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