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Properties

Label 1-65e2-4225.29-r0-0-0
Degree $1$
Conductor $4225$
Sign $0.998 - 0.0510i$
Analytic cond. $19.6208$
Root an. cond. $19.6208$
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  + (−0.619 + 0.784i)2-s + (−0.999 + 0.0322i)3-s + (−0.231 − 0.972i)4-s + (0.594 − 0.804i)6-s + (0.919 − 0.391i)7-s + (0.906 + 0.421i)8-s + (0.997 − 0.0643i)9-s + (0.997 + 0.0643i)11-s + (0.262 + 0.964i)12-s + (−0.262 + 0.964i)14-s + (−0.892 + 0.450i)16-s + (−0.0884 − 0.996i)17-s + (−0.568 + 0.822i)18-s + (−0.978 + 0.207i)19-s + (−0.906 + 0.421i)21-s + (−0.669 + 0.743i)22-s + ⋯
L(s)  = 1  + (−0.619 + 0.784i)2-s + (−0.999 + 0.0322i)3-s + (−0.231 − 0.972i)4-s + (0.594 − 0.804i)6-s + (0.919 − 0.391i)7-s + (0.906 + 0.421i)8-s + (0.997 − 0.0643i)9-s + (0.997 + 0.0643i)11-s + (0.262 + 0.964i)12-s + (−0.262 + 0.964i)14-s + (−0.892 + 0.450i)16-s + (−0.0884 − 0.996i)17-s + (−0.568 + 0.822i)18-s + (−0.978 + 0.207i)19-s + (−0.906 + 0.421i)21-s + (−0.669 + 0.743i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0510i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0510i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4225\)    =    \(5^{2} \cdot 13^{2}\)
Sign: $0.998 - 0.0510i$
Analytic conductor: \(19.6208\)
Root analytic conductor: \(19.6208\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4225} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4225,\ (0:\ ),\ 0.998 - 0.0510i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8439402661 + 0.02154989716i\)
\(L(\frac12)\) \(\approx\) \(0.8439402661 + 0.02154989716i\)
\(L(1)\) \(\approx\) \(0.6306325660 + 0.1312577632i\)
\(L(1)\) \(\approx\) \(0.6306325660 + 0.1312577632i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.619 + 0.784i)T \)
3 \( 1 + (-0.999 + 0.0322i)T \)
7 \( 1 + (0.919 - 0.391i)T \)
11 \( 1 + (0.997 + 0.0643i)T \)
17 \( 1 + (-0.0884 - 0.996i)T \)
19 \( 1 + (-0.978 + 0.207i)T \)
23 \( 1 + (-0.913 + 0.406i)T \)
29 \( 1 + (-0.962 + 0.270i)T \)
31 \( 1 + (0.399 + 0.916i)T \)
37 \( 1 + (-0.737 - 0.675i)T \)
41 \( 1 + (0.339 + 0.940i)T \)
43 \( 1 + (0.200 - 0.979i)T \)
47 \( 1 + (0.607 - 0.794i)T \)
53 \( 1 + (-0.981 + 0.192i)T \)
59 \( 1 + (-0.704 + 0.709i)T \)
61 \( 1 + (-0.999 + 0.0161i)T \)
67 \( 1 + (0.231 - 0.972i)T \)
71 \( 1 + (-0.471 + 0.881i)T \)
73 \( 1 + (0.443 + 0.896i)T \)
79 \( 1 + (-0.607 + 0.794i)T \)
83 \( 1 + (0.527 - 0.849i)T \)
89 \( 1 + (0.913 - 0.406i)T \)
97 \( 1 + (0.932 + 0.362i)T \)
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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

−18.33453130220556618378293910057, −17.6541486914925959564035526441, −17.12111129628721663267981661016, −16.85034647859356213259810249445, −15.8280846474071553450734142020, −15.10837955863629606652821219940, −14.274776835341782159160051011746, −13.38236683534289525234587567218, −12.52487634066886435957554467363, −12.16677343977851118223968739277, −11.3964905074668849376951829555, −10.97402680925268768021397737684, −10.34124187392217771333108494069, −9.50240351416468021596747975732, −8.81049638003819556775760020975, −8.04768601573150823237937355937, −7.41948252476647552246578339216, −6.33856924070560692437179601432, −5.92815175332265103046650833181, −4.5979215504890545640035585521, −4.35028565213784415521766869021, −3.439206583174625243662870778104, −1.99905484133732486422902589052, −1.796051580138185735004406633453, −0.724196199677491861078901323879, 0.491696092204992294924006674155, 1.44683200957198781190648938957, 1.99498655647936110256826689338, 3.7685367421024722195141404211, 4.47661403280344384780094224912, 5.07924593110405445144202020148, 5.854239280198084503549484820362, 6.51655353062647079351736833040, 7.23882173657493334206908493016, 7.71434768704934262505361201607, 8.7414660533536276843621515135, 9.303564193679113180160281208213, 10.22882582036476836600532415864, 10.69592767301019170907794534011, 11.46945210323175825256861220421, 11.958905306964974345667789496845, 12.94303751111591716113123550763, 13.973985090282747311392760850325, 14.28749302523595372463868427134, 15.204398995631496766592054026424, 15.79082558722616738955840843635, 16.54726439927267639539640449783, 17.14643080628715987512629594039, 17.442646072583256945588177070933, 18.26715155204527571644017614585

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