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Properties

Label 1-47-47.38-r1-0-0
Degree $1$
Conductor $47$
Sign $-0.0324 - 0.999i$
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  + (−0.576 − 0.816i)2-s + (−0.775 + 0.631i)3-s + (−0.334 + 0.942i)4-s + (−0.682 + 0.730i)5-s + (0.962 + 0.269i)6-s + (0.460 − 0.887i)7-s + (0.962 − 0.269i)8-s + (0.203 − 0.979i)9-s + (0.990 + 0.136i)10-s + (−0.854 − 0.519i)11-s + (−0.334 − 0.942i)12-s + (0.917 + 0.398i)13-s + (−0.990 + 0.136i)14-s + (0.0682 − 0.997i)15-s + (−0.775 − 0.631i)16-s + (0.854 − 0.519i)17-s + ⋯
L(s)  = 1  + (−0.576 − 0.816i)2-s + (−0.775 + 0.631i)3-s + (−0.334 + 0.942i)4-s + (−0.682 + 0.730i)5-s + (0.962 + 0.269i)6-s + (0.460 − 0.887i)7-s + (0.962 − 0.269i)8-s + (0.203 − 0.979i)9-s + (0.990 + 0.136i)10-s + (−0.854 − 0.519i)11-s + (−0.334 − 0.942i)12-s + (0.917 + 0.398i)13-s + (−0.990 + 0.136i)14-s + (0.0682 − 0.997i)15-s + (−0.775 − 0.631i)16-s + (0.854 − 0.519i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0324 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0324 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(47\)
Sign: $-0.0324 - 0.999i$
Analytic conductor: \(5.05085\)
Root analytic conductor: \(5.05085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{47} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 47,\ (1:\ ),\ -0.0324 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4195915484 - 0.4334397724i\)
\(L(\frac12)\) \(\approx\) \(0.4195915484 - 0.4334397724i\)
\(L(1)\) \(\approx\) \(0.5468630817 - 0.1720936290i\)
\(L(1)\) \(\approx\) \(0.5468630817 - 0.1720936290i\)

Euler product

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

−34.3631419110917441116113896498, −33.323327768807563826679707539588, −31.85248929224610718569698075962, −30.81383819408653003177874818395, −29.01885243043104120956419800379, −27.961447271708338198786750673476, −27.61339353547339180779865607068, −25.62959760144887601984582721179, −24.72475528893080422044723826495, −23.591077843245280173349213158960, −23.0360738321421919406872099644, −21.02913867529842072658963677171, −19.284754005962722015487050920636, −18.38445523425798128826512416719, −17.30934227731332723383486042081, −16.10250684145117312191773238022, −15.13625901406258686806327085976, −13.18879177081730128236822840517, −11.95106123104995074839401109361, −10.47707453644762737043840513041, −8.50423811687190747030597072201, −7.703071202302681868757861087292, −5.94455183250554462380539425033, −4.922665916326753456615718229098, −1.361457912743473374120328834304, 0.54844235831134229978497198098, 3.27829111682910326260411236614, 4.58113554388575990079144732279, 6.929622398694795983857455141568, 8.46396822734974166201659661393, 10.4050172316572243401913788062, 10.87982644097360398021158629676, 12.01330536400815598366580754577, 13.812891247646196323893895566919, 15.677693002464083568115777723669, 16.79245332600929130631662011292, 18.04120556427953450899148489023, 19.059323831512387368050254931066, 20.640293380233054806466349139472, 21.42439584180208155236554605483, 22.88048675071611330701598494281, 23.57182994255073265748841877959, 26.068703537093850426393953032925, 26.80810063136099535897846148970, 27.63779662886647611662972565193, 28.789898063865794968109532892510, 29.91254722539064390177560542305, 30.8430316388916171804235744333, 32.29262717807194268259707340510, 33.92905322285915807395429906873

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