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Properties

Label 1-47-47.10-r1-0-0
Degree $1$
Conductor $47$
Sign $-0.714 - 0.699i$
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  + (−0.917 + 0.398i)2-s + (−0.0682 + 0.997i)3-s + (0.682 − 0.730i)4-s + (−0.854 + 0.519i)5-s + (−0.334 − 0.942i)6-s + (0.203 + 0.979i)7-s + (−0.334 + 0.942i)8-s + (−0.990 − 0.136i)9-s + (0.576 − 0.816i)10-s + (0.775 − 0.631i)11-s + (0.682 + 0.730i)12-s + (−0.962 − 0.269i)13-s + (−0.576 − 0.816i)14-s + (−0.460 − 0.887i)15-s + (−0.0682 − 0.997i)16-s + (−0.775 − 0.631i)17-s + ⋯
L(s)  = 1  + (−0.917 + 0.398i)2-s + (−0.0682 + 0.997i)3-s + (0.682 − 0.730i)4-s + (−0.854 + 0.519i)5-s + (−0.334 − 0.942i)6-s + (0.203 + 0.979i)7-s + (−0.334 + 0.942i)8-s + (−0.990 − 0.136i)9-s + (0.576 − 0.816i)10-s + (0.775 − 0.631i)11-s + (0.682 + 0.730i)12-s + (−0.962 − 0.269i)13-s + (−0.576 − 0.816i)14-s + (−0.460 − 0.887i)15-s + (−0.0682 − 0.997i)16-s + (−0.775 − 0.631i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.714 - 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.714 - 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(47\)
Sign: $-0.714 - 0.699i$
Analytic conductor: \(5.05085\)
Root analytic conductor: \(5.05085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{47} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 47,\ (1:\ ),\ -0.714 - 0.699i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1135762170 + 0.2784196479i\)
\(L(\frac12)\) \(\approx\) \(-0.1135762170 + 0.2784196479i\)
\(L(1)\) \(\approx\) \(0.3750170130 + 0.3100761498i\)
\(L(1)\) \(\approx\) \(0.3750170130 + 0.3100761498i\)

Euler product

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

−33.398352337115230947074133608444, −31.49739823697555441205912720219, −30.45971537087582004993291749720, −29.61719457211758421475721496741, −28.44532380053027562497867762001, −27.3833124255925160587092088097, −26.27339920995309430123778540346, −24.833155766593295006034179380425, −23.975964100448772097360346065955, −22.62271056474130424263645511911, −20.6007206056125267235877382669, −19.72475483425169692871112610892, −18.99777316365227044177416201607, −17.26499213280095852321487998177, −16.89364014984045982579229294412, −14.89153902374280906263986383116, −12.96014312363982413269639158176, −12.01656469826791101797862082190, −10.84696943758436775126568685493, −8.99786490452106930319038362126, −7.70975844568035543403873789287, −6.85607468781438264787841826540, −4.07716730582679223225458845347, −1.83046530017871812543218201596, −0.23408529944540239693464183976, 2.884769344282960895639571203544, 5.00349745579655056395374308887, 6.67594297506599811061578450793, 8.40386204985630538837450864026, 9.37556631687425510954884797285, 10.94738604246051312937087404502, 11.72470790934502343516237946145, 14.67375457673813869602505099641, 15.25522977071763100184111291268, 16.370971803433253671992831015070, 17.63375569030629551879442835720, 19.08835725879370180134124271025, 19.95220945240262342320768147803, 21.56990053456855863951463454470, 22.68644130443732896714292839601, 24.23834112356588598235915945460, 25.4240201625381391338837820092, 26.78565460871594092958885749815, 27.31622417196971824065150305624, 28.236768157234460675515563504306, 29.609541791101396552945585651138, 31.35535790605584471448531385819, 32.340388337863069295421813895629, 33.72145697057060412101915620842, 34.55563525707135773648235966466

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