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Properties

Label 1-389-389.74-r0-0-0
Degree $1$
Conductor $389$
Sign $-0.799 - 0.601i$
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  + (0.966 − 0.256i)2-s + (0.452 − 0.891i)3-s + (0.868 − 0.495i)4-s + (−0.974 − 0.224i)5-s + (0.208 − 0.977i)6-s + (−0.986 + 0.161i)7-s + (0.712 − 0.701i)8-s + (−0.590 − 0.807i)9-s + (−0.999 + 0.0323i)10-s + (0.834 − 0.550i)11-s + (−0.0485 − 0.998i)12-s + (−0.423 − 0.905i)13-s + (−0.912 + 0.408i)14-s + (−0.641 + 0.767i)15-s + (0.509 − 0.860i)16-s + (−0.999 + 0.0323i)17-s + ⋯
L(s)  = 1  + (0.966 − 0.256i)2-s + (0.452 − 0.891i)3-s + (0.868 − 0.495i)4-s + (−0.974 − 0.224i)5-s + (0.208 − 0.977i)6-s + (−0.986 + 0.161i)7-s + (0.712 − 0.701i)8-s + (−0.590 − 0.807i)9-s + (−0.999 + 0.0323i)10-s + (0.834 − 0.550i)11-s + (−0.0485 − 0.998i)12-s + (−0.423 − 0.905i)13-s + (−0.912 + 0.408i)14-s + (−0.641 + 0.767i)15-s + (0.509 − 0.860i)16-s + (−0.999 + 0.0323i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.799 - 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.799 - 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(389\)
Sign: $-0.799 - 0.601i$
Analytic conductor: \(1.80650\)
Root analytic conductor: \(1.80650\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{389} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 389,\ (0:\ ),\ -0.799 - 0.601i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5958627267 - 1.783519305i\)
\(L(\frac12)\) \(\approx\) \(0.5958627267 - 1.783519305i\)
\(L(1)\) \(\approx\) \(1.238980082 - 0.9992313558i\)
\(L(1)\) \(\approx\) \(1.238980082 - 0.9992313558i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad389 \( 1 \)
good2 \( 1 + (0.966 - 0.256i)T \)
3 \( 1 + (0.452 - 0.891i)T \)
5 \( 1 + (-0.974 - 0.224i)T \)
7 \( 1 + (-0.986 + 0.161i)T \)
11 \( 1 + (0.834 - 0.550i)T \)
13 \( 1 + (-0.423 - 0.905i)T \)
17 \( 1 + (-0.999 + 0.0323i)T \)
19 \( 1 + (0.333 + 0.942i)T \)
23 \( 1 + (-0.974 + 0.224i)T \)
29 \( 1 + (0.925 + 0.378i)T \)
31 \( 1 + (0.0161 - 0.999i)T \)
37 \( 1 + (-0.302 - 0.953i)T \)
41 \( 1 + (0.333 - 0.942i)T \)
43 \( 1 + (-0.363 + 0.931i)T \)
47 \( 1 + (0.333 - 0.942i)T \)
53 \( 1 + (0.665 + 0.746i)T \)
59 \( 1 + (0.145 + 0.989i)T \)
61 \( 1 + (-0.423 - 0.905i)T \)
67 \( 1 + (0.797 - 0.603i)T \)
71 \( 1 + (0.868 - 0.495i)T \)
73 \( 1 + (0.509 + 0.860i)T \)
79 \( 1 + (0.0161 + 0.999i)T \)
83 \( 1 + (0.797 + 0.603i)T \)
89 \( 1 + (0.393 - 0.919i)T \)
97 \( 1 + (0.868 + 0.495i)T \)
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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

−24.75561480530305670404636339589, −23.811488639744080825657119794715, −22.8772943886409179495943984, −22.18473559268176080134804412686, −21.74665952660799537653683313006, −20.337340222323261851504817958989, −19.823656112535229677674157163, −19.27730734067482522452228284252, −17.43085913546720782262203572112, −16.37012382914433716458828355805, −15.85105731892718246126214403077, −15.14045986922033355411691939102, −14.26604487107066929606944483243, −13.4819001084277471042976342346, −12.20467469658527232855757544647, −11.5524294219424675193755966743, −10.49012588895941217807788756291, −9.38152529974230691895020790886, −8.34144169634672336531175083507, −7.051716062278812198392336314185, −6.49154817868633831635951628162, −4.77273406826633506888653628213, −4.21797362819397170098126192035, −3.350252242870577586443170315284, −2.36402062247171364696111161446, 0.747810503394679614620158316414, 2.29071945123425088581783850410, 3.37540389144609953730808949598, 3.9970809502100531304460839584, 5.63229200185676266918388351657, 6.49234703370301260387721635646, 7.387234025035198879614160083006, 8.385825139739122353861467735485, 9.5886627620623114292409193454, 10.92534624516370713588807572818, 12.08038935655137291333462818298, 12.357882728550213921738397284092, 13.34115557299897773915414832596, 14.16416518460400024062896546104, 15.15730336606253909715750703354, 15.88478400780124044980619352952, 16.87352431447533230707768299748, 18.33798221663820461620821122883, 19.30079510140512379379803660611, 19.85419503756132001472155810928, 20.25409411149481884505441002715, 21.69933078861433953497483991971, 22.71322684816055837406761355384, 23.04152028536796169563748291394, 24.3265795376682918025386621625

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