↑

Properties

Label 1-640-640.563-r0-0-0
Degree $1$
Conductor $640$
Sign $0.0136 - 0.999i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.555 − 0.831i)3-s + (0.923 − 0.382i)7-s + (−0.382 + 0.923i)9-s + (0.555 − 0.831i)11-s + (0.195 − 0.980i)13-s + (0.707 − 0.707i)17-s + (−0.195 + 0.980i)19-s + (−0.831 − 0.555i)21-s + (−0.382 + 0.923i)23-s + (0.980 − 0.195i)27-s + (−0.555 − 0.831i)29-s + i·31-s − 33-s + (0.980 − 0.195i)37-s + (−0.923 + 0.382i)39-s + ⋯
L(s)  = 1  + (−0.555 − 0.831i)3-s + (0.923 − 0.382i)7-s + (−0.382 + 0.923i)9-s + (0.555 − 0.831i)11-s + (0.195 − 0.980i)13-s + (0.707 − 0.707i)17-s + (−0.195 + 0.980i)19-s + (−0.831 − 0.555i)21-s + (−0.382 + 0.923i)23-s + (0.980 − 0.195i)27-s + (−0.555 − 0.831i)29-s + i·31-s − 33-s + (0.980 − 0.195i)37-s + (−0.923 + 0.382i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0136 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0136 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $0.0136 - 0.999i$
Analytic conductor: \(2.97214\)
Root analytic conductor: \(2.97214\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (563, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 640,\ (0:\ ),\ 0.0136 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9141070072 - 0.9266363322i\)
\(L(\frac12)\) \(\approx\) \(0.9141070072 - 0.9266363322i\)
\(L(1)\) \(\approx\) \(0.9464606316 - 0.4117328885i\)
\(L(1)\) \(\approx\) \(0.9464606316 - 0.4117328885i\)

Euler product

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

−23.08039183077494191669966148655, −22.10399555225361512163660881345, −21.5847550124268195099504695619, −20.779864272084618195667021085824, −20.08662127374634297913195844125, −18.894392250554918610494425023179, −17.99932305490925525636828125733, −17.24997289632298461277227962107, −16.59427232464073875900040855675, −15.63373350244927767194027300192, −14.66416690759691224729961098821, −14.433456931926593392956669272824, −12.84817842753868893024702998396, −11.9549290330941748473924619645, −11.30114824413144697418371840948, −10.5119689149664216213072915387, −9.41787886863991512176099860082, −8.85350874891046683430335406109, −7.63279218643156185369748886522, −6.48400800102714588980545763091, −5.6371336499774916643121203905, −4.48788911105573619032811332945, −4.13927225338549078643298783360, −2.54576679914219624781427872932, −1.31983341562441837130911146453, 0.81023328688734578251262935730, 1.66977917642051591538606239648, 3.04553156239020350932857847373, 4.26037246061665781928094892543, 5.571911683123475259147444446409, 5.936810712462453631601982437846, 7.400630747111586069252763976, 7.811612185987765808848771035171, 8.80133762762431980228138872702, 10.19412947251461619614302911785, 11.01148402523787331847155574747, 11.746877865840111245444760392995, 12.47694303872493735531497185843, 13.65654437541834614661935463695, 14.04780767894344515028663863096, 15.13370949335071993509193453083, 16.38184392345229004564656396898, 16.95908478798568715643201707594, 17.85370662235062583072644054622, 18.41403201196424428155064611665, 19.33951785094116055007972493962, 20.15652924626699379690329469151, 21.095048127866108235689299842546, 21.9417694159201868506125555101, 23.01121570309792814310078175815

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