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Properties

Label 1-47-47.16-r0-0-0
Degree $1$
Conductor $47$
Sign $-0.648 + 0.761i$
Analytic cond. $0.218267$
Root an. cond. $0.218267$
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  + (0.460 + 0.887i)2-s + (−0.334 + 0.942i)3-s + (−0.576 + 0.816i)4-s + (−0.917 − 0.398i)5-s + (−0.990 + 0.136i)6-s + (0.854 + 0.519i)7-s + (−0.990 − 0.136i)8-s + (−0.775 − 0.631i)9-s + (−0.0682 − 0.997i)10-s + (0.962 − 0.269i)11-s + (−0.576 − 0.816i)12-s + (0.203 + 0.979i)13-s + (−0.0682 + 0.997i)14-s + (0.682 − 0.730i)15-s + (−0.334 − 0.942i)16-s + (0.962 + 0.269i)17-s + ⋯
L(s)  = 1  + (0.460 + 0.887i)2-s + (−0.334 + 0.942i)3-s + (−0.576 + 0.816i)4-s + (−0.917 − 0.398i)5-s + (−0.990 + 0.136i)6-s + (0.854 + 0.519i)7-s + (−0.990 − 0.136i)8-s + (−0.775 − 0.631i)9-s + (−0.0682 − 0.997i)10-s + (0.962 − 0.269i)11-s + (−0.576 − 0.816i)12-s + (0.203 + 0.979i)13-s + (−0.0682 + 0.997i)14-s + (0.682 − 0.730i)15-s + (−0.334 − 0.942i)16-s + (0.962 + 0.269i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.648 + 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.648 + 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(47\)
Sign: $-0.648 + 0.761i$
Analytic conductor: \(0.218267\)
Root analytic conductor: \(0.218267\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{47} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 47,\ (0:\ ),\ -0.648 + 0.761i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3524137752 + 0.7625681811i\)
\(L(\frac12)\) \(\approx\) \(0.3524137752 + 0.7625681811i\)
\(L(1)\) \(\approx\) \(0.6979500436 + 0.7082085153i\)
\(L(1)\) \(\approx\) \(0.6979500436 + 0.7082085153i\)

Euler product

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

−33.697735414050095723153839585563, −32.2087488381057822437519031726, −30.831691046835642577520924277283, −30.3009914820051998976179330477, −29.49730099010676760648394281332, −27.75924556210127656841215602271, −27.39221884945625930378470369561, −25.22216253147050992309899013442, −23.71058974019313621812080175481, −23.2645452863246808773237310248, −22.0574713377215197869096718601, −20.33647633325687673026744201371, −19.502770755432134422977945856870, −18.38480295353606201874030498035, −17.23083754769652926735561995974, −15.005167340303639356126148920279, −13.96895870448406250632545607477, −12.51999728253769192748483286858, −11.56113239526317376462632052413, −10.61633483606369504238664475729, −8.35938219994104436158008414145, −6.90958843918949674696598215782, −5.05810899834039297200635612400, −3.359678932149474118346833022230, −1.31970846488547369754495390964, 3.8189810745566361127376845760, 4.73483244390191792415978199850, 6.21888920849293170699353931106, 8.15031178134269207439726764681, 9.126231267003787962144206625863, 11.36409462267435549062929648547, 12.25195921912506173927757392476, 14.406423409379940565708256383135, 15.1050288295724726427878642635, 16.424273407839426540935250827, 17.08081060332350072427127777197, 18.87101869888595869384500155001, 20.79224306826036249170372610376, 21.6343961665352642526782491823, 22.92073676217148944113841955022, 23.8503516681168894608703624958, 25.012594970070853090102644053501, 26.52256275271858619872122706369, 27.42385394443949331849986851640, 28.18250481484366315070145849394, 30.318028465657692677192122895681, 31.48950456682215237792727334369, 32.209905266244705302075334110488, 33.39539195755684364414471719687, 34.37340125009889508230039502624

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