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Properties

Label 1-4235-4235.179-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.874 - 0.485i$
Analytic cond. $19.6672$
Root an. cond. $19.6672$
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  + (0.999 − 0.0190i)2-s + (−0.913 − 0.406i)3-s + (0.999 − 0.0380i)4-s + (−0.921 − 0.389i)6-s + (0.998 − 0.0570i)8-s + (0.669 + 0.743i)9-s + (−0.928 − 0.371i)12-s + (0.985 + 0.170i)13-s + (0.997 − 0.0760i)16-s + (0.398 + 0.917i)17-s + (0.683 + 0.730i)18-s + (0.861 − 0.508i)19-s + (−0.235 + 0.971i)23-s + (−0.935 − 0.353i)24-s + (0.988 + 0.151i)26-s + (−0.309 − 0.951i)27-s + ⋯
L(s)  = 1  + (0.999 − 0.0190i)2-s + (−0.913 − 0.406i)3-s + (0.999 − 0.0380i)4-s + (−0.921 − 0.389i)6-s + (0.998 − 0.0570i)8-s + (0.669 + 0.743i)9-s + (−0.928 − 0.371i)12-s + (0.985 + 0.170i)13-s + (0.997 − 0.0760i)16-s + (0.398 + 0.917i)17-s + (0.683 + 0.730i)18-s + (0.861 − 0.508i)19-s + (−0.235 + 0.971i)23-s + (−0.935 − 0.353i)24-s + (0.988 + 0.151i)26-s + (−0.309 − 0.951i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.874 - 0.485i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.874 - 0.485i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.006327131 - 0.7794248065i\)
\(L(\frac12)\) \(\approx\) \(3.006327131 - 0.7794248065i\)
\(L(1)\) \(\approx\) \(1.721161719 - 0.2350301506i\)
\(L(1)\) \(\approx\) \(1.721161719 - 0.2350301506i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.999 - 0.0190i)T \)
3 \( 1 + (-0.913 - 0.406i)T \)
13 \( 1 + (0.985 + 0.170i)T \)
17 \( 1 + (0.398 + 0.917i)T \)
19 \( 1 + (0.861 - 0.508i)T \)
23 \( 1 + (-0.235 + 0.971i)T \)
29 \( 1 + (0.198 - 0.980i)T \)
31 \( 1 + (-0.0665 - 0.997i)T \)
37 \( 1 + (-0.272 - 0.962i)T \)
41 \( 1 + (0.0855 - 0.996i)T \)
43 \( 1 + (-0.841 - 0.540i)T \)
47 \( 1 + (0.683 - 0.730i)T \)
53 \( 1 + (-0.997 - 0.0760i)T \)
59 \( 1 + (0.820 - 0.572i)T \)
61 \( 1 + (0.483 + 0.875i)T \)
67 \( 1 + (-0.981 - 0.189i)T \)
71 \( 1 + (-0.736 + 0.676i)T \)
73 \( 1 + (0.179 + 0.983i)T \)
79 \( 1 + (0.964 + 0.263i)T \)
83 \( 1 + (0.0285 + 0.999i)T \)
89 \( 1 + (0.580 + 0.814i)T \)
97 \( 1 + (-0.774 + 0.633i)T \)
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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

−18.27587867765065660598387233839, −17.78603766212333737156662152398, −16.701578793963559878165727396656, −16.214251771382806705412885682354, −15.93477084192662513253451948254, −15.03160636010486951015655922423, −14.356224851748947126498023135837, −13.68906600914604864057169035086, −12.9025635727576472522142238008, −12.22112485736602470452550375981, −11.72198886942245000852121490663, −11.02950983795729075581762459081, −10.41298301085624876142829664907, −9.78458727733484671654337278583, −8.74480782656658061897560209107, −7.780866683507793450301025176427, −6.97658435118907655796394194455, −6.30562422709075344678047955326, −5.75133193510914989509128950837, −4.88274788639461372337579757521, −4.56452879456664287780543098431, −3.334299453540939358511838429453, −3.155296400162205022595288287700, −1.64974535617638364616654347636, −0.96746168171776961001723355954, 0.83437362245775026525176228653, 1.6749185177687477044636450414, 2.40705404314767232463161074483, 3.67610626808288840895684571713, 4.01308201891034371021092981722, 5.11986918370354477504281509194, 5.671656383469399158865347983801, 6.17012561871703158994492814337, 6.9951287650732463141772491360, 7.59756777981707799590896125919, 8.371402928671460808552989116231, 9.59763422961153154436613394123, 10.383222462506864665091727590144, 11.03930284625192163798613941653, 11.639411345560428918136885326038, 12.11668384971995019537508267022, 12.96070929616638545653615868511, 13.49033480065329663040449712351, 13.964791361033786154399711529515, 15.00339179881005739779354235051, 15.69387735846810697080700168031, 16.109277886215102664015751997977, 16.963954761583525913165306668910, 17.4687408435027227907918852973, 18.29394195892991599537946854147

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