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Properties

Label 1-800-800.291-r1-0-0
Degree $1$
Conductor $800$
Sign $-0.701 + 0.712i$
Analytic cond. $85.9719$
Root an. cond. $85.9719$
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  + (0.987 + 0.156i)3-s + i·7-s + (0.951 + 0.309i)9-s + (−0.891 − 0.453i)11-s + (−0.891 + 0.453i)13-s + (0.809 + 0.587i)17-s + (−0.156 − 0.987i)19-s + (−0.156 + 0.987i)21-s + (0.951 − 0.309i)23-s + (0.891 + 0.453i)27-s + (−0.987 − 0.156i)29-s + (0.809 + 0.587i)31-s + (−0.809 − 0.587i)33-s + (0.453 + 0.891i)37-s + (−0.951 + 0.309i)39-s + ⋯
L(s)  = 1  + (0.987 + 0.156i)3-s + i·7-s + (0.951 + 0.309i)9-s + (−0.891 − 0.453i)11-s + (−0.891 + 0.453i)13-s + (0.809 + 0.587i)17-s + (−0.156 − 0.987i)19-s + (−0.156 + 0.987i)21-s + (0.951 − 0.309i)23-s + (0.891 + 0.453i)27-s + (−0.987 − 0.156i)29-s + (0.809 + 0.587i)31-s + (−0.809 − 0.587i)33-s + (0.453 + 0.891i)37-s + (−0.951 + 0.309i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $-0.701 + 0.712i$
Analytic conductor: \(85.9719\)
Root analytic conductor: \(85.9719\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (291, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 800,\ (1:\ ),\ -0.701 + 0.712i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7464921487 + 1.782362048i\)
\(L(\frac12)\) \(\approx\) \(0.7464921487 + 1.782362048i\)
\(L(1)\) \(\approx\) \(1.251721173 + 0.3906640038i\)
\(L(1)\) \(\approx\) \(1.251721173 + 0.3906640038i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.987 + 0.156i)T \)
7 \( 1 + iT \)
11 \( 1 + (-0.891 - 0.453i)T \)
13 \( 1 + (-0.891 + 0.453i)T \)
17 \( 1 + (0.809 + 0.587i)T \)
19 \( 1 + (-0.156 - 0.987i)T \)
23 \( 1 + (0.951 - 0.309i)T \)
29 \( 1 + (-0.987 - 0.156i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (0.453 + 0.891i)T \)
41 \( 1 + (0.951 + 0.309i)T \)
43 \( 1 + (-0.707 + 0.707i)T \)
47 \( 1 + (-0.809 + 0.587i)T \)
53 \( 1 + (-0.156 + 0.987i)T \)
59 \( 1 + (-0.453 - 0.891i)T \)
61 \( 1 + (-0.453 + 0.891i)T \)
67 \( 1 + (0.156 + 0.987i)T \)
71 \( 1 + (-0.587 - 0.809i)T \)
73 \( 1 + (-0.951 + 0.309i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (-0.156 - 0.987i)T \)
89 \( 1 + (0.951 - 0.309i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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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

−21.4778161484365581396711097103, −20.74376697910249491355273533260, −20.31644761865825044303693047151, −19.40253970345504701389335821968, −18.72241892870540842740192294937, −17.83891607423425887656210576607, −16.90428615956657366559063183182, −16.08732993893453736137212961689, −15.013551494295731244995503795098, −14.54883133011570554551787782475, −13.57976739993667136864996690732, −12.95103797914240610727171438477, −12.16659662067966964794637674584, −10.828216333373694665099449397288, −9.98737684315222620581721796619, −9.49795354154156103043638484270, −8.13611350675591768922798715979, −7.57287360924768369769708504877, −6.999319174043168925879805854015, −5.50545909429877054449918631027, −4.52451767944025455176686903264, −3.53099269824390028510208655878, −2.682621791496765050965717762995, −1.5909330467461724158192634114, −0.3486392522875184911114722065, 1.41356235057810443650605030398, 2.67611349292471126551005503324, 2.97871683959657227693375962225, 4.46654937974332388851299970929, 5.2127879981782059167985739221, 6.38385948937156274403408769794, 7.51021228231663783284634988289, 8.24024029374186456984864302414, 9.06230137621355039890378272584, 9.74867781262748840954337665578, 10.72271605842698516211375559969, 11.76478406461653715802321910840, 12.79208708831308737438528819643, 13.29914027986294812763645845187, 14.49360406780929330007576877567, 14.957914225035726504902972622610, 15.73483306784855604904797998211, 16.55706709607317501523872630236, 17.63479374264269175642768556756, 18.77612642854995034336865608558, 19.00317521418460136436086422287, 19.893766035212707925974538934170, 20.94616900646999740000029460449, 21.46751452117813260303329845661, 22.016258120795444167140984642483

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