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Properties

Label 1-640-640.323-r0-0-0
Degree $1$
Conductor $640$
Sign $0.395 + 0.918i$
Analytic cond. $2.97214$
Root an. cond. $2.97214$
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  + (−0.980 − 0.195i)3-s + (0.382 + 0.923i)7-s + (0.923 + 0.382i)9-s + (0.980 − 0.195i)11-s + (0.555 + 0.831i)13-s + (−0.707 + 0.707i)17-s + (−0.555 − 0.831i)19-s + (−0.195 − 0.980i)21-s + (0.923 + 0.382i)23-s + (−0.831 − 0.555i)27-s + (−0.980 − 0.195i)29-s + i·31-s − 33-s + (−0.831 − 0.555i)37-s + (−0.382 − 0.923i)39-s + ⋯
L(s)  = 1  + (−0.980 − 0.195i)3-s + (0.382 + 0.923i)7-s + (0.923 + 0.382i)9-s + (0.980 − 0.195i)11-s + (0.555 + 0.831i)13-s + (−0.707 + 0.707i)17-s + (−0.555 − 0.831i)19-s + (−0.195 − 0.980i)21-s + (0.923 + 0.382i)23-s + (−0.831 − 0.555i)27-s + (−0.980 − 0.195i)29-s + i·31-s − 33-s + (−0.831 − 0.555i)37-s + (−0.382 − 0.923i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.395 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.395 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $0.395 + 0.918i$
Analytic conductor: \(2.97214\)
Root analytic conductor: \(2.97214\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 640,\ (0:\ ),\ 0.395 + 0.918i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8526447797 + 0.5613622453i\)
\(L(\frac12)\) \(\approx\) \(0.8526447797 + 0.5613622453i\)
\(L(1)\) \(\approx\) \(0.8548874858 + 0.1607377622i\)
\(L(1)\) \(\approx\) \(0.8548874858 + 0.1607377622i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.980 - 0.195i)T \)
7 \( 1 + (0.382 + 0.923i)T \)
11 \( 1 + (0.980 - 0.195i)T \)
13 \( 1 + (0.555 + 0.831i)T \)
17 \( 1 + (-0.707 + 0.707i)T \)
19 \( 1 + (-0.555 - 0.831i)T \)
23 \( 1 + (0.923 + 0.382i)T \)
29 \( 1 + (-0.980 - 0.195i)T \)
31 \( 1 + iT \)
37 \( 1 + (-0.831 - 0.555i)T \)
41 \( 1 + (0.382 - 0.923i)T \)
43 \( 1 + (0.980 - 0.195i)T \)
47 \( 1 + (-0.707 + 0.707i)T \)
53 \( 1 + (0.195 + 0.980i)T \)
59 \( 1 + (0.831 + 0.555i)T \)
61 \( 1 + (-0.195 + 0.980i)T \)
67 \( 1 + (-0.980 - 0.195i)T \)
71 \( 1 + (0.923 - 0.382i)T \)
73 \( 1 + (-0.382 + 0.923i)T \)
79 \( 1 + (0.707 - 0.707i)T \)
83 \( 1 + (0.831 - 0.555i)T \)
89 \( 1 + (0.382 + 0.923i)T \)
97 \( 1 - T \)
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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

−22.68964717029514691501106396624, −22.289005920914814746312177955467, −20.9277549923468962072446010889, −20.58212183517094049240045449719, −19.479934405456594058540018087703, −18.42438905584727536335126801582, −17.671290274851056189632422128969, −16.95063232481745674772602940265, −16.40195073257526516575792032034, −15.287796056330947110069977871162, −14.53844497905648387283105309584, −13.37258567796090238387902366445, −12.68918460850915656360318829086, −11.53327646260215915125014318584, −11.02519020913942317910593539850, −10.18082881211223417804954112202, −9.26905549182176714348391919592, −8.04202918241221651781222777402, −7.00517616265512038843213733976, −6.32268769632915440044365083521, −5.21788531669991297660650764874, −4.321484641827263116300811933190, −3.53153726754899528551582188958, −1.730629975415072519134215555500, −0.6559665136859129766592418819, 1.28535993487645726382321088252, 2.19182760698968471158661027307, 3.83334233764008519291322225546, 4.73687275651864205147250342414, 5.75857980025603462844138645224, 6.478357218302562882832485124694, 7.30668374528191881717503255687, 8.80374452942159314338943755415, 9.15659568148264184662687224786, 10.736324796249668002225408408117, 11.23239287084631090548788230784, 12.026479180644255871928341238261, 12.81079330820655695341327476448, 13.77476422156821311378910942688, 14.88470781016244883199817758772, 15.64470911489437321395607061708, 16.53282418065336734325448479363, 17.42283045767669482114446013180, 17.893590557961924911458147411700, 19.10346829064504236518657422782, 19.32260218881889947173456444935, 20.92730597150846064117054206799, 21.571297217201728876504573276469, 22.19694301677916991099241049762, 22.9935348045121450345055303854

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