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Properties

Label 1-320-320.123-r0-0-0
Degree $1$
Conductor $320$
Sign $0.661 - 0.750i$
Analytic cond. $1.48607$
Root an. cond. $1.48607$
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  + (0.923 − 0.382i)3-s + (0.707 − 0.707i)7-s + (0.707 − 0.707i)9-s + (0.382 − 0.923i)11-s + (0.382 + 0.923i)13-s − 17-s + (−0.923 + 0.382i)19-s + (0.382 − 0.923i)21-s + (−0.707 − 0.707i)23-s + (0.382 − 0.923i)27-s + (0.382 + 0.923i)29-s + 31-s − i·33-s + (−0.382 + 0.923i)37-s + (0.707 + 0.707i)39-s + ⋯
L(s)  = 1  + (0.923 − 0.382i)3-s + (0.707 − 0.707i)7-s + (0.707 − 0.707i)9-s + (0.382 − 0.923i)11-s + (0.382 + 0.923i)13-s − 17-s + (−0.923 + 0.382i)19-s + (0.382 − 0.923i)21-s + (−0.707 − 0.707i)23-s + (0.382 − 0.923i)27-s + (0.382 + 0.923i)29-s + 31-s − i·33-s + (−0.382 + 0.923i)37-s + (0.707 + 0.707i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.661 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.661 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.661 - 0.750i$
Analytic conductor: \(1.48607\)
Root analytic conductor: \(1.48607\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 320,\ (0:\ ),\ 0.661 - 0.750i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.687975559 - 0.7620167389i\)
\(L(\frac12)\) \(\approx\) \(1.687975559 - 0.7620167389i\)
\(L(1)\) \(\approx\) \(1.453398221 - 0.3617091325i\)
\(L(1)\) \(\approx\) \(1.453398221 - 0.3617091325i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.923 - 0.382i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
11 \( 1 + (0.382 - 0.923i)T \)
13 \( 1 + (0.382 + 0.923i)T \)
17 \( 1 - T \)
19 \( 1 + (-0.923 + 0.382i)T \)
23 \( 1 + (-0.707 - 0.707i)T \)
29 \( 1 + (0.382 + 0.923i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.382 + 0.923i)T \)
41 \( 1 + (0.707 - 0.707i)T \)
43 \( 1 + (-0.923 - 0.382i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.923 - 0.382i)T \)
59 \( 1 + (0.923 + 0.382i)T \)
61 \( 1 + (0.382 + 0.923i)T \)
67 \( 1 + (-0.923 + 0.382i)T \)
71 \( 1 + (0.707 + 0.707i)T \)
73 \( 1 + (-0.707 - 0.707i)T \)
79 \( 1 - iT \)
83 \( 1 + (0.382 + 0.923i)T \)
89 \( 1 + (-0.707 - 0.707i)T \)
97 \( 1 - iT \)
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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

−25.09400046904666719772119305704, −24.800099290603053131841111374398, −23.54337402695520387826454249728, −22.40376153746975775135370555806, −21.57913042207762423945770026055, −20.776078257425546807929535649990, −19.9684344623627560263594617429, −19.194401119882469605948101142298, −17.9765296895736302131392183216, −17.38779193324803011752031960842, −15.724666138343026021721201735737, −15.335153743460358547859121862662, −14.4914064202794326249731395571, −13.458500696819236355495045426069, −12.53110176363677146021750559982, −11.35140900447923446782388736376, −10.294955787254162842818733545367, −9.29606510163329620355512362473, −8.42947053550950269476821275880, −7.65319640737817878800319294764, −6.27551962402809108084137730616, −4.88664156784245545224672435051, −4.07564291546609441050866096360, −2.65197269069618147047916482039, −1.795622875920625049205882727751, 1.23318659517157454608667131168, 2.3265519566872601940226068124, 3.75666436556559878791400134906, 4.49048118219766593098606986607, 6.30279105273205113107059126483, 7.05571276343611273113573829656, 8.425515318124549444474390588852, 8.69084424539441712626974478837, 10.15060623117789805627494847397, 11.14483212229941358138560523344, 12.19264545040348812408373127275, 13.455719795650570679698009958047, 13.99214371597519443717346835442, 14.74622004909045723420029076184, 15.92077695652355614772720531476, 16.92371558384825686903886496643, 17.95021036468359767150598683734, 18.89481936288996382034568038447, 19.61469377063868445240081945107, 20.576384343826418839287694632117, 21.220140950942384099060931659065, 22.23019793431429095704246587389, 23.747387955725816820966316681442, 24.007680370515795768769799065296, 24.966255224193207728430360531701

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