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Properties

Label 1-229-229.9-r0-0-0
Degree $1$
Conductor $229$
Sign $0.239 - 0.971i$
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  + (−0.401 + 0.915i)2-s + (−0.998 − 0.0550i)3-s + (−0.677 − 0.735i)4-s + (0.350 − 0.936i)5-s + (0.451 − 0.892i)6-s + (0.137 + 0.990i)7-s + (0.945 − 0.324i)8-s + (0.993 + 0.110i)9-s + (0.716 + 0.697i)10-s + (−0.879 − 0.475i)11-s + (0.635 + 0.771i)12-s + (−0.986 + 0.164i)13-s + (−0.962 − 0.272i)14-s + (−0.401 + 0.915i)15-s + (−0.0825 + 0.996i)16-s + (−0.986 + 0.164i)17-s + ⋯
L(s)  = 1  + (−0.401 + 0.915i)2-s + (−0.998 − 0.0550i)3-s + (−0.677 − 0.735i)4-s + (0.350 − 0.936i)5-s + (0.451 − 0.892i)6-s + (0.137 + 0.990i)7-s + (0.945 − 0.324i)8-s + (0.993 + 0.110i)9-s + (0.716 + 0.697i)10-s + (−0.879 − 0.475i)11-s + (0.635 + 0.771i)12-s + (−0.986 + 0.164i)13-s + (−0.962 − 0.272i)14-s + (−0.401 + 0.915i)15-s + (−0.0825 + 0.996i)16-s + (−0.986 + 0.164i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.239 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.239 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(229\)
Sign: $0.239 - 0.971i$
Analytic conductor: \(1.06347\)
Root analytic conductor: \(1.06347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{229} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 229,\ (0:\ ),\ 0.239 - 0.971i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2801489034 - 0.2195541892i\)
\(L(\frac12)\) \(\approx\) \(0.2801489034 - 0.2195541892i\)
\(L(1)\) \(\approx\) \(0.5171571460 + 0.05247688871i\)
\(L(1)\) \(\approx\) \(0.5171571460 + 0.05247688871i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad229 \( 1 \)
good2 \( 1 + (-0.401 + 0.915i)T \)
3 \( 1 + (-0.998 - 0.0550i)T \)
5 \( 1 + (0.350 - 0.936i)T \)
7 \( 1 + (0.137 + 0.990i)T \)
11 \( 1 + (-0.879 - 0.475i)T \)
13 \( 1 + (-0.986 + 0.164i)T \)
17 \( 1 + (-0.986 + 0.164i)T \)
19 \( 1 + (0.635 - 0.771i)T \)
23 \( 1 + (0.716 - 0.697i)T \)
29 \( 1 + (-0.926 - 0.376i)T \)
31 \( 1 + (0.851 - 0.523i)T \)
37 \( 1 + (-0.821 + 0.569i)T \)
41 \( 1 + (-0.592 - 0.805i)T \)
43 \( 1 + (-0.0825 - 0.996i)T \)
47 \( 1 + (0.993 + 0.110i)T \)
53 \( 1 + (0.546 - 0.837i)T \)
59 \( 1 + (-0.821 - 0.569i)T \)
61 \( 1 + (-0.401 - 0.915i)T \)
67 \( 1 + (-0.592 + 0.805i)T \)
71 \( 1 + (0.0275 - 0.999i)T \)
73 \( 1 + (-0.754 - 0.656i)T \)
79 \( 1 + (-0.926 + 0.376i)T \)
83 \( 1 + (0.904 - 0.426i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.191 - 0.981i)T \)
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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

−26.751465785683748234235305930525, −26.17389863308490625527057576821, −24.6883231315840751394083498655, −23.260259459680722699932516945020, −22.798191923081125104529069680479, −21.90776069918150023240692991599, −21.06581975326723814486661747703, −20.0473333283608533644741737479, −18.93285539383763661401020002439, −17.97043113659528106652659696895, −17.51194801385804338224475674399, −16.57504881353588513969410201383, −15.211156013744611602914823524158, −13.826056610321744969859943877931, −12.98387253122811595854404824690, −11.86152382415018845780350304801, −10.87235496189935937964348887844, −10.33115256059220154401896554217, −9.55262663396949837001077876008, −7.58222166177847882915486737741, −7.04949066661554070682369962720, −5.352642489452280805779834909228, −4.30002677844819987385598059911, −2.931558902324480371999543774044, −1.53882702332800621541698555484, 0.33880651180002593070628240868, 2.05940717735309961266627021491, 4.70044471389651480436178549300, 5.21280722021711406885385972035, 6.10349985709445188359839663507, 7.28414672624101793005169114943, 8.54099193183911350174329155055, 9.37395401192747768317337983683, 10.469411091818934554814756638568, 11.73360927850137195621204895503, 12.82750602980881472911070326319, 13.628321745390887749489770835156, 15.298410607025687693304355938778, 15.77342112530185082920848012480, 16.89328611871499787247811524853, 17.4342862233207084002422929863, 18.399075089324017154293109565760, 19.18413268313825817492828198267, 20.69280000364140316666180653536, 21.88725525281111411688333228474, 22.436293877668399029967233042329, 23.870716225462803848932151165965, 24.31802201944349184716456128591, 24.87920694377590001756651947465, 26.23325345549973664032865327537

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