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Properties

Label 1-47-47.43-r1-0-0
Degree $1$
Conductor $47$
Sign $0.387 + 0.921i$
Analytic cond. $5.05085$
Root an. cond. $5.05085$
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  + (0.854 + 0.519i)2-s + (−0.576 − 0.816i)3-s + (0.460 + 0.887i)4-s + (−0.203 + 0.979i)5-s + (−0.0682 − 0.997i)6-s + (0.962 + 0.269i)7-s + (−0.0682 + 0.997i)8-s + (−0.334 + 0.942i)9-s + (−0.682 + 0.730i)10-s + (0.990 − 0.136i)11-s + (0.460 − 0.887i)12-s + (0.775 + 0.631i)13-s + (0.682 + 0.730i)14-s + (0.917 − 0.398i)15-s + (−0.576 + 0.816i)16-s + (−0.990 − 0.136i)17-s + ⋯
L(s)  = 1  + (0.854 + 0.519i)2-s + (−0.576 − 0.816i)3-s + (0.460 + 0.887i)4-s + (−0.203 + 0.979i)5-s + (−0.0682 − 0.997i)6-s + (0.962 + 0.269i)7-s + (−0.0682 + 0.997i)8-s + (−0.334 + 0.942i)9-s + (−0.682 + 0.730i)10-s + (0.990 − 0.136i)11-s + (0.460 − 0.887i)12-s + (0.775 + 0.631i)13-s + (0.682 + 0.730i)14-s + (0.917 − 0.398i)15-s + (−0.576 + 0.816i)16-s + (−0.990 − 0.136i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.387 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.387 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(47\)
Sign: $0.387 + 0.921i$
Analytic conductor: \(5.05085\)
Root analytic conductor: \(5.05085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{47} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 47,\ (1:\ ),\ 0.387 + 0.921i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.774073123 + 1.178087415i\)
\(L(\frac12)\) \(\approx\) \(1.774073123 + 1.178087415i\)
\(L(1)\) \(\approx\) \(1.453806486 + 0.5456874278i\)
\(L(1)\) \(\approx\) \(1.453806486 + 0.5456874278i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad47 \( 1 \)
good2 \( 1 + (0.854 + 0.519i)T \)
3 \( 1 + (-0.576 - 0.816i)T \)
5 \( 1 + (-0.203 + 0.979i)T \)
7 \( 1 + (0.962 + 0.269i)T \)
11 \( 1 + (0.990 - 0.136i)T \)
13 \( 1 + (0.775 + 0.631i)T \)
17 \( 1 + (-0.990 - 0.136i)T \)
19 \( 1 + (-0.203 - 0.979i)T \)
23 \( 1 + (-0.854 + 0.519i)T \)
29 \( 1 + (0.775 - 0.631i)T \)
31 \( 1 + (0.576 - 0.816i)T \)
37 \( 1 + (0.682 - 0.730i)T \)
41 \( 1 + (0.0682 + 0.997i)T \)
43 \( 1 + (-0.460 - 0.887i)T \)
53 \( 1 + (-0.0682 - 0.997i)T \)
59 \( 1 + (0.460 - 0.887i)T \)
61 \( 1 + (0.682 + 0.730i)T \)
67 \( 1 + (-0.962 + 0.269i)T \)
71 \( 1 + (0.854 - 0.519i)T \)
73 \( 1 + (0.334 + 0.942i)T \)
79 \( 1 + (-0.917 + 0.398i)T \)
83 \( 1 + (-0.990 + 0.136i)T \)
89 \( 1 + (0.203 - 0.979i)T \)
97 \( 1 + (-0.576 - 0.816i)T \)
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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

−33.08169384764291194802707060337, −32.65374626552233037077911998102, −31.36458300126529962418888346113, −30.21480305809247681188736127256, −28.863763716160446392188782700872, −27.89127775710427376256444474830, −27.21737852000941760616356183591, −24.999417553095972833546858075879, −23.89776784151949429952273902627, −22.909638560906363460976946648758, −21.69501041870522835082857385690, −20.65380818278196110726399911045, −19.97994254254906980352835912206, −17.82826684297660508628318711251, −16.47629893028945021959108280850, −15.32844441001303341856595229501, −14.07220347088770469626001497893, −12.42550666075231115115583178709, −11.45977568502027822578674501056, −10.28828925202471956895671968487, −8.65023933205781092736699738838, −6.16003645183172222459952551930, −4.78314966558417845620060959540, −3.934566656345212608813008217180, −1.22142252152402355643058008214, 2.21009059772691473219152755579, 4.30312520167874989333969317866, 6.077664589655561494018793703779, 6.961547093594566851593080518142, 8.32927368247460186043533088719, 11.326620568641899879630053674745, 11.61602135185125259970601119664, 13.46710157127900053876295210544, 14.37041906337960323377571531856, 15.69525790244830223474245372240, 17.29804182632152544051516966473, 18.16030949638591400942001101327, 19.68181394819776243927228527948, 21.558140804224573250297921830356, 22.39666314234579847994927837125, 23.5477832082557950712159081113, 24.37078862213131930101002337776, 25.505350662016022962682058638344, 26.861525763927145282844311851108, 28.40340992368253761313018858895, 30.08189488364629926211395640663, 30.3822880502253551174824755156, 31.42760141076957221310757090434, 33.24745832129502038815189526869, 33.94926220825467197116134970840

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