Properties

Label 1-389-389.77-r0-0-0
Degree $1$
Conductor $389$
Sign $0.615 - 0.788i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.240 + 0.970i)2-s + (0.145 + 0.989i)3-s + (−0.884 − 0.466i)4-s + (0.0161 + 0.999i)5-s + (−0.995 − 0.0970i)6-s + (−0.423 − 0.905i)7-s + (0.665 − 0.746i)8-s + (−0.957 + 0.287i)9-s + (−0.974 − 0.224i)10-s + (−0.590 − 0.807i)11-s + (0.333 − 0.942i)12-s + (0.0808 + 0.996i)13-s + (0.981 − 0.193i)14-s + (−0.986 + 0.161i)15-s + (0.563 + 0.825i)16-s + (−0.974 − 0.224i)17-s + ⋯
L(s)  = 1  + (−0.240 + 0.970i)2-s + (0.145 + 0.989i)3-s + (−0.884 − 0.466i)4-s + (0.0161 + 0.999i)5-s + (−0.995 − 0.0970i)6-s + (−0.423 − 0.905i)7-s + (0.665 − 0.746i)8-s + (−0.957 + 0.287i)9-s + (−0.974 − 0.224i)10-s + (−0.590 − 0.807i)11-s + (0.333 − 0.942i)12-s + (0.0808 + 0.996i)13-s + (0.981 − 0.193i)14-s + (−0.986 + 0.161i)15-s + (0.563 + 0.825i)16-s + (−0.974 − 0.224i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(389\)
Sign: $0.615 - 0.788i$
Analytic conductor: \(1.80650\)
Root analytic conductor: \(1.80650\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{389} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 389,\ (0:\ ),\ 0.615 - 0.788i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06647537586 - 0.03244055800i\)
\(L(\frac12)\) \(\approx\) \(0.06647537586 - 0.03244055800i\)
\(L(1)\) \(\approx\) \(0.4485427407 + 0.3961823478i\)
\(L(1)\) \(\approx\) \(0.4485427407 + 0.3961823478i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad389 \( 1 \)
good2 \( 1 + (-0.240 + 0.970i)T \)
3 \( 1 + (0.145 + 0.989i)T \)
5 \( 1 + (0.0161 + 0.999i)T \)
7 \( 1 + (-0.423 - 0.905i)T \)
11 \( 1 + (-0.590 - 0.807i)T \)
13 \( 1 + (0.0808 + 0.996i)T \)
17 \( 1 + (-0.974 - 0.224i)T \)
19 \( 1 + (-0.689 - 0.724i)T \)
23 \( 1 + (0.0161 - 0.999i)T \)
29 \( 1 + (-0.912 - 0.408i)T \)
31 \( 1 + (-0.113 - 0.993i)T \)
37 \( 1 + (0.834 + 0.550i)T \)
41 \( 1 + (-0.689 + 0.724i)T \)
43 \( 1 + (0.509 - 0.860i)T \)
47 \( 1 + (-0.689 + 0.724i)T \)
53 \( 1 + (0.925 + 0.378i)T \)
59 \( 1 + (-0.852 + 0.523i)T \)
61 \( 1 + (0.0808 + 0.996i)T \)
67 \( 1 + (-0.177 - 0.984i)T \)
71 \( 1 + (-0.884 - 0.466i)T \)
73 \( 1 + (0.563 - 0.825i)T \)
79 \( 1 + (-0.113 + 0.993i)T \)
83 \( 1 + (-0.177 + 0.984i)T \)
89 \( 1 + (-0.302 + 0.953i)T \)
97 \( 1 + (-0.884 + 0.466i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.771112346901569243980365868467, −23.54903218667212132279781822482, −22.93281014728051914726970577225, −21.85389804361012286530666365764, −20.88789410687331227016042367966, −20.04157796810600562197481782162, −19.56177854885472526516488341872, −18.48257091379003282940800054741, −17.84077413068555348229126720462, −17.11513798336363667569281409927, −15.80821404059236212902607786812, −14.69425044502882226288470837141, −13.21542941554652850258223209815, −12.89581750634332843202508858715, −12.30296315298350472430887914119, −11.33294773322230343801993430005, −10.05542050615130973987448905509, −9.01670022333275690108716577816, −8.39404338176144793433780227005, −7.46837143774167255396834876076, −5.86105858172266418261540419928, −5.01844714257378420010200096501, −3.54518928654498890213194320696, −2.331066755853237151809536973577, −1.580709182743911356964028402255, 0.04497536097164828883635943464, 2.5934606775203136596168965998, 3.89812462914204425024790384319, 4.574198554788695082481771317382, 6.02913753666035441105686858278, 6.71301622444552121565822453147, 7.79611002507499663368142178830, 8.891375826351048535152947694467, 9.738862639140948205753452761985, 10.68686583401086313428843587396, 11.20341388834921479529863844249, 13.38429555397013708409205672428, 13.797120198714304866495737994612, 14.849850142796070271803826238014, 15.44851037613391813125024401080, 16.49863960859076259260727436806, 16.904260471432448924223135014289, 18.12903757474387061291775494662, 19.00813543802256657082694832902, 19.79943218386487135108899892231, 21.04371973139640060335201354337, 22.07301534216543502552780335889, 22.541886063321057490372555429760, 23.52101935472551656813260628380, 24.23777958446152805097379183793

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