Properties

Label 1-229-229.37-r0-0-0
Degree $1$
Conductor $229$
Sign $0.787 + 0.615i$
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  + (−0.0825 + 0.996i)2-s + (−0.298 − 0.954i)3-s + (−0.986 − 0.164i)4-s + (−0.926 − 0.376i)5-s + (0.975 − 0.218i)6-s + (0.0275 + 0.999i)7-s + (0.245 − 0.969i)8-s + (−0.821 + 0.569i)9-s + (0.451 − 0.892i)10-s + (−0.401 + 0.915i)11-s + (0.137 + 0.990i)12-s + (0.789 − 0.614i)13-s + (−0.998 − 0.0550i)14-s + (−0.0825 + 0.996i)15-s + (0.945 + 0.324i)16-s + (0.789 − 0.614i)17-s + ⋯
L(s)  = 1  + (−0.0825 + 0.996i)2-s + (−0.298 − 0.954i)3-s + (−0.986 − 0.164i)4-s + (−0.926 − 0.376i)5-s + (0.975 − 0.218i)6-s + (0.0275 + 0.999i)7-s + (0.245 − 0.969i)8-s + (−0.821 + 0.569i)9-s + (0.451 − 0.892i)10-s + (−0.401 + 0.915i)11-s + (0.137 + 0.990i)12-s + (0.789 − 0.614i)13-s + (−0.998 − 0.0550i)14-s + (−0.0825 + 0.996i)15-s + (0.945 + 0.324i)16-s + (0.789 − 0.614i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.787 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.787 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(229\)
Sign: $0.787 + 0.615i$
Analytic conductor: \(1.06347\)
Root analytic conductor: \(1.06347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{229} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 229,\ (0:\ ),\ 0.787 + 0.615i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7152175034 + 0.2464188663i\)
\(L(\frac12)\) \(\approx\) \(0.7152175034 + 0.2464188663i\)
\(L(1)\) \(\approx\) \(0.7307257420 + 0.1760485147i\)
\(L(1)\) \(\approx\) \(0.7307257420 + 0.1760485147i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad229 \( 1 \)
good2 \( 1 + (-0.0825 + 0.996i)T \)
3 \( 1 + (-0.298 - 0.954i)T \)
5 \( 1 + (-0.926 - 0.376i)T \)
7 \( 1 + (0.0275 + 0.999i)T \)
11 \( 1 + (-0.401 + 0.915i)T \)
13 \( 1 + (0.789 - 0.614i)T \)
17 \( 1 + (0.789 - 0.614i)T \)
19 \( 1 + (0.137 - 0.990i)T \)
23 \( 1 + (0.451 + 0.892i)T \)
29 \( 1 + (0.851 - 0.523i)T \)
31 \( 1 + (0.993 - 0.110i)T \)
37 \( 1 + (-0.191 + 0.981i)T \)
41 \( 1 + (0.904 - 0.426i)T \)
43 \( 1 + (0.945 - 0.324i)T \)
47 \( 1 + (-0.821 + 0.569i)T \)
53 \( 1 + (-0.677 + 0.735i)T \)
59 \( 1 + (-0.191 - 0.981i)T \)
61 \( 1 + (-0.0825 - 0.996i)T \)
67 \( 1 + (0.904 + 0.426i)T \)
71 \( 1 + (-0.592 + 0.805i)T \)
73 \( 1 + (0.716 + 0.697i)T \)
79 \( 1 + (0.851 + 0.523i)T \)
83 \( 1 + (-0.754 + 0.656i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.962 - 0.272i)T \)
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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

−26.627774518902676548217312524321, −26.01043164039090423407269585456, −23.89802175254154247997964318022, −23.102172523183250239970699232678, −22.67545198215836141386007945475, −21.21264236943250972410590777568, −20.976708207431252941727346779986, −19.75446836623435179555407543232, −19.03454174690164376913391642001, −17.961730760303604305249249229911, −16.63590282572205552083758183290, −16.1624947689347474980830862663, −14.60638862700247540565369487977, −13.964729070294773454568147732649, −12.52062219073536955908244326036, −11.46228452202574276950598545605, −10.73322695756753899343087070480, −10.2018353742690671079180041404, −8.74891659631091338463153123372, −7.91593403155154542010695019898, −6.158506277527823418036001619247, −4.654279996307514653112279412450, −3.79954403111044231114022857937, −3.125158229193207838714876895641, −0.87566100324863521575936493685, 0.979441293388065647053598270221, 2.95762280779515828438570693345, 4.75836732952980209658921277890, 5.557342256402458633529969931371, 6.75276589843484980811074023129, 7.758875254298179795570620567748, 8.363858930229936856551165174522, 9.53342076635548991700676420304, 11.25151921523161966221903296009, 12.312069879383387035216588528879, 12.96807259547731209782553925173, 14.0998932477965555044039079066, 15.56063970314787186818455382992, 15.66546441467398576113204316935, 17.17803206130970936258834661820, 17.92543148628466229327514626187, 18.77788725102333315311729122165, 19.502252945465833556140246350198, 20.83700697076422628758042573649, 22.37154402772206648313568994008, 23.11784036871022554845010565804, 23.670020474106986067633577901540, 24.71104496953178153475514754519, 25.27232433615904274868953023341, 26.14351169082655400919877519930

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