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Properties

Label 1-320-320.83-r0-0-0
Degree $1$
Conductor $320$
Sign $0.966 - 0.256i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.966 - 0.256i$
Analytic conductor: \(1.48607\)
Root analytic conductor: \(1.48607\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 320,\ (0:\ ),\ 0.966 - 0.256i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.658946804 - 0.2160848773i\)
\(L(\frac12)\) \(\approx\) \(1.658946804 - 0.2160848773i\)
\(L(1)\) \(\approx\) \(1.378868392 - 0.04443667409i\)
\(L(1)\) \(\approx\) \(1.378868392 - 0.04443667409i\)

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

−25.31669981241807477194087821625, −24.46122113729877038141247377104, −23.4525904688222059866877547345, −22.60247432857544447248239953108, −21.35265269181466736973219251186, −20.773126788014788465661490776311, −19.68851773399772421499367919477, −18.9145497960168438329011953952, −18.326082477195607972482893767665, −17.11676818679805541235843442873, −15.783457612986332998179770357805, −15.28903851054857426617388870133, −14.15165753298957586209629425081, −13.327366199894494484838498036195, −12.428469112091853111633341533554, −11.59940641243074253395570767984, −9.835155520702246413718276601753, −9.452018362578098057411322808041, −8.29284740949144838887813905150, −7.29403926270249213212577135408, −6.38870708012402180774354000647, −5.01814044209831860114196101862, −3.608065946783097238804006150, −2.6870300523373399804122542749, −1.525977078075829570013000641716, 1.13409003873585916473606865418, 3.101916791856905410649865726898, 3.34269332860763137523502768566, 4.83532372023616873439811232836, 6.058412243755649375373187303125, 7.44264936055883530486750206896, 8.16537593334265381112596924241, 9.27617793600917991721354917079, 10.22044753285701959836647146033, 10.89011018819076750144135075597, 12.51669878570751097518088888663, 13.37503890534471309695017675617, 14.082889409375861891934220770276, 15.07122847940759992809161994763, 16.18681520654655580223277482048, 16.55276146459171044365956654752, 18.11766795402135949215822897352, 19.00647406220530608921134553150, 19.79392027026885714170357269627, 20.64570383387495384462161903281, 21.30875951700223578276780877835, 22.48344756456754248335880945441, 23.20465354096718895618879423738, 24.44260658420386051739304127865, 25.17134715302388853336334861740

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