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Properties

Label 1-229-229.14-r0-0-0
Degree $1$
Conductor $229$
Sign $-0.759 - 0.650i$
Analytic cond. $1.06347$
Root an. cond. $1.06347$
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  + (0.245 − 0.969i)2-s + (0.137 − 0.990i)3-s + (−0.879 − 0.475i)4-s + (0.993 + 0.110i)5-s + (−0.926 − 0.376i)6-s + (0.904 + 0.426i)7-s + (−0.677 + 0.735i)8-s + (−0.962 − 0.272i)9-s + (0.350 − 0.936i)10-s + (0.945 − 0.324i)11-s + (−0.592 + 0.805i)12-s + (−0.401 − 0.915i)13-s + (0.635 − 0.771i)14-s + (0.245 − 0.969i)15-s + (0.546 + 0.837i)16-s + (−0.401 − 0.915i)17-s + ⋯
L(s)  = 1  + (0.245 − 0.969i)2-s + (0.137 − 0.990i)3-s + (−0.879 − 0.475i)4-s + (0.993 + 0.110i)5-s + (−0.926 − 0.376i)6-s + (0.904 + 0.426i)7-s + (−0.677 + 0.735i)8-s + (−0.962 − 0.272i)9-s + (0.350 − 0.936i)10-s + (0.945 − 0.324i)11-s + (−0.592 + 0.805i)12-s + (−0.401 − 0.915i)13-s + (0.635 − 0.771i)14-s + (0.245 − 0.969i)15-s + (0.546 + 0.837i)16-s + (−0.401 − 0.915i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.759 - 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.759 - 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(229\)
Sign: $-0.759 - 0.650i$
Analytic conductor: \(1.06347\)
Root analytic conductor: \(1.06347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{229} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 229,\ (0:\ ),\ -0.759 - 0.650i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5425813081 - 1.466637424i\)
\(L(\frac12)\) \(\approx\) \(0.5425813081 - 1.466637424i\)
\(L(1)\) \(\approx\) \(0.9117271501 - 1.008549753i\)
\(L(1)\) \(\approx\) \(0.9117271501 - 1.008549753i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad229 \( 1 \)
good2 \( 1 + (0.245 - 0.969i)T \)
3 \( 1 + (0.137 - 0.990i)T \)
5 \( 1 + (0.993 + 0.110i)T \)
7 \( 1 + (0.904 + 0.426i)T \)
11 \( 1 + (0.945 - 0.324i)T \)
13 \( 1 + (-0.401 - 0.915i)T \)
17 \( 1 + (-0.401 - 0.915i)T \)
19 \( 1 + (-0.592 - 0.805i)T \)
23 \( 1 + (0.350 + 0.936i)T \)
29 \( 1 + (-0.821 + 0.569i)T \)
31 \( 1 + (-0.191 + 0.981i)T \)
37 \( 1 + (-0.998 - 0.0550i)T \)
41 \( 1 + (0.716 + 0.697i)T \)
43 \( 1 + (0.546 - 0.837i)T \)
47 \( 1 + (-0.962 - 0.272i)T \)
53 \( 1 + (0.789 + 0.614i)T \)
59 \( 1 + (-0.998 + 0.0550i)T \)
61 \( 1 + (0.245 + 0.969i)T \)
67 \( 1 + (0.716 - 0.697i)T \)
71 \( 1 + (-0.754 + 0.656i)T \)
73 \( 1 + (0.975 + 0.218i)T \)
79 \( 1 + (-0.821 - 0.569i)T \)
83 \( 1 + (0.451 - 0.892i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.298 + 0.954i)T \)
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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

−26.38548735828718707067054475029, −26.01655703339254486262831378944, −24.78558624395687969327762847440, −24.25270044167999344575823713497, −22.88626031529812215493306318126, −22.12585309352579276663310178020, −21.2574526522306221027157661718, −20.67940516277041831898359810825, −19.16465544143218179215365213018, −17.74096141265753533455467840585, −16.950557414042507653622199682612, −16.66057913970358922973615905928, −15.04628721972912157267108080633, −14.50987978376584473331371066763, −13.868756127647973091815041040128, −12.54402967843965288202076664178, −11.111699462382875211036685566163, −9.90791631562326780518995499223, −9.091062997822767441907993491639, −8.19411931642024170172875332277, −6.71514534544185482128066043635, −5.73524603627964750662765173622, −4.56819495035385994172779959034, −3.97201249810011127950548947697, −1.98559708944313989780238738664, 1.192051256181818967935659602465, 2.14583631648886501876522325632, 3.12940703003654546146441804523, 4.99621735038882995993679742733, 5.80102553791060688533436188068, 7.1396770385370969443840148227, 8.67771632428333215851921580157, 9.26978872998215954694972636626, 10.77394557209246613860398746272, 11.58159779872600073113927725258, 12.568027162765536231458755133801, 13.47355198215035911395959891677, 14.21659485651861140316431188470, 15.00303150159026268366233416393, 17.24946290857891264337338053471, 17.73338645316211482402252527466, 18.48188834239343507234694144027, 19.556995312257534103782828865823, 20.32577219126389959731458859991, 21.38328685901345156504982382936, 22.1405295172198651355403317435, 23.07167362690198472305444012502, 24.34608317994401606482643270612, 24.787333436249807679082764042338, 25.88764413855249854125125956853

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