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Properties

Label 1-47-47.45-r1-0-0
Degree $1$
Conductor $47$
Sign $0.882 - 0.471i$
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  + (0.962 + 0.269i)2-s + (0.460 − 0.887i)3-s + (0.854 + 0.519i)4-s + (0.775 − 0.631i)5-s + (0.682 − 0.730i)6-s + (−0.990 − 0.136i)7-s + (0.682 + 0.730i)8-s + (−0.576 − 0.816i)9-s + (0.917 − 0.398i)10-s + (0.0682 + 0.997i)11-s + (0.854 − 0.519i)12-s + (0.334 − 0.942i)13-s + (−0.917 − 0.398i)14-s + (−0.203 − 0.979i)15-s + (0.460 + 0.887i)16-s + (−0.0682 + 0.997i)17-s + ⋯
L(s)  = 1  + (0.962 + 0.269i)2-s + (0.460 − 0.887i)3-s + (0.854 + 0.519i)4-s + (0.775 − 0.631i)5-s + (0.682 − 0.730i)6-s + (−0.990 − 0.136i)7-s + (0.682 + 0.730i)8-s + (−0.576 − 0.816i)9-s + (0.917 − 0.398i)10-s + (0.0682 + 0.997i)11-s + (0.854 − 0.519i)12-s + (0.334 − 0.942i)13-s + (−0.917 − 0.398i)14-s + (−0.203 − 0.979i)15-s + (0.460 + 0.887i)16-s + (−0.0682 + 0.997i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.882 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.882 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(47\)
Sign: $0.882 - 0.471i$
Analytic conductor: \(5.05085\)
Root analytic conductor: \(5.05085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{47} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 47,\ (1:\ ),\ 0.882 - 0.471i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.973126758 - 0.7443158689i\)
\(L(\frac12)\) \(\approx\) \(2.973126758 - 0.7443158689i\)
\(L(1)\) \(\approx\) \(2.114569287 - 0.3245773575i\)
\(L(1)\) \(\approx\) \(2.114569287 - 0.3245773575i\)

Euler product

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

−33.50747292119810368396053326982, −32.58534030229172693328472474791, −31.80338091887091646657507923882, −30.60571806013506270134936383345, −29.314449973653882210295844952099, −28.49631475343833093906484629542, −26.63471359842476412153954869241, −25.738872170935572454184250831700, −24.59573222597253757902409982889, −22.826885638727520461808039636197, −21.96513369608255026820992208435, −21.25629943145182649263797246042, −19.89994692451246749613191022265, −18.74060639550259546068192993917, −16.45960639327466722694542781746, −15.63968160662902239602928507266, −14.0435575406126761855537055576, −13.63671943447985132325987743843, −11.59104223571229618677150610595, −10.32519743633432249152873478125, −9.23901036206221894799084149988, −6.72527943715888996699858099512, −5.445891477968143358432966283537, −3.645056905771525671567278668782, −2.56676050570884590063715934869, 1.81964463385389177670149217266, 3.49851940414197814366535624003, 5.593241268191388743535423643798, 6.712423965782865511175221287624, 8.18357649080427178578582124766, 9.971007982852661817700855657560, 12.28529424096519627798438730971, 12.93286935897334046595553030858, 13.88622593845225912793971801137, 15.282462528267585062700233161649, 16.772451159846052217887454261802, 17.99042752707767473862707213787, 19.89769043771815379349166883765, 20.502155351024653920295190791838, 22.122681986874552204541219634, 23.238622937324919604575735850920, 24.379449659435865085559512047228, 25.47393077226584363518569480205, 25.86809045891245627717427558071, 28.43787886359855590161218601173, 29.41896992461087074016856390959, 30.299582337993661339659825270149, 31.49847027274767792826428278629, 32.494794744182445807967776498524, 33.29712345843366342114857679694

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