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Properties

Label 1-232-232.165-r0-0-0
Degree $1$
Conductor $232$
Sign $0.995 - 0.0954i$
Analytic cond. $1.07740$
Root an. cond. $1.07740$
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  + (0.222 − 0.974i)3-s + (−0.623 + 0.781i)5-s + (−0.222 + 0.974i)7-s + (−0.900 − 0.433i)9-s + (0.900 − 0.433i)11-s + (0.900 − 0.433i)13-s + (0.623 + 0.781i)15-s + 17-s + (0.222 + 0.974i)19-s + (0.900 + 0.433i)21-s + (0.623 + 0.781i)23-s + (−0.222 − 0.974i)25-s + (−0.623 + 0.781i)27-s + (0.623 − 0.781i)31-s + (−0.222 − 0.974i)33-s + ⋯
L(s)  = 1  + (0.222 − 0.974i)3-s + (−0.623 + 0.781i)5-s + (−0.222 + 0.974i)7-s + (−0.900 − 0.433i)9-s + (0.900 − 0.433i)11-s + (0.900 − 0.433i)13-s + (0.623 + 0.781i)15-s + 17-s + (0.222 + 0.974i)19-s + (0.900 + 0.433i)21-s + (0.623 + 0.781i)23-s + (−0.222 − 0.974i)25-s + (−0.623 + 0.781i)27-s + (0.623 − 0.781i)31-s + (−0.222 − 0.974i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(232\)    =    \(2^{3} \cdot 29\)
Sign: $0.995 - 0.0954i$
Analytic conductor: \(1.07740\)
Root analytic conductor: \(1.07740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{232} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 232,\ (0:\ ),\ 0.995 - 0.0954i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.201049024 - 0.05742126850i\)
\(L(\frac12)\) \(\approx\) \(1.201049024 - 0.05742126850i\)
\(L(1)\) \(\approx\) \(1.081299369 - 0.08903198484i\)
\(L(1)\) \(\approx\) \(1.081299369 - 0.08903198484i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 \)
good3 \( 1 + (0.222 - 0.974i)T \)
5 \( 1 + (-0.623 + 0.781i)T \)
7 \( 1 + (-0.222 + 0.974i)T \)
11 \( 1 + (0.900 - 0.433i)T \)
13 \( 1 + (0.900 - 0.433i)T \)
17 \( 1 + T \)
19 \( 1 + (0.222 + 0.974i)T \)
23 \( 1 + (0.623 + 0.781i)T \)
31 \( 1 + (0.623 - 0.781i)T \)
37 \( 1 + (0.900 + 0.433i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.623 - 0.781i)T \)
47 \( 1 + (-0.900 + 0.433i)T \)
53 \( 1 + (-0.623 + 0.781i)T \)
59 \( 1 - T \)
61 \( 1 + (0.222 - 0.974i)T \)
67 \( 1 + (0.900 + 0.433i)T \)
71 \( 1 + (-0.900 + 0.433i)T \)
73 \( 1 + (0.623 + 0.781i)T \)
79 \( 1 + (-0.900 - 0.433i)T \)
83 \( 1 + (0.222 + 0.974i)T \)
89 \( 1 + (0.623 - 0.781i)T \)
97 \( 1 + (-0.222 - 0.974i)T \)
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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

−26.46472988000040052128441969414, −25.56595114450083150351084064248, −24.52946346557370213814039506726, −23.24659952730356553826440904206, −22.88394430651813590049699841209, −21.48374807476389802892263384346, −20.68978876151007272441954233641, −19.93517409657909169202856307893, −19.27268906984042417671048048583, −17.57137257621023766312727957348, −16.54445174553829094724627651972, −16.211624349744352232738827998866, −14.98787769948996573949843714773, −14.07701808730230168465991024666, −12.99319311339349050077370107926, −11.71512177813747082138495796499, −10.86537008295985447222541271398, −9.67364128331359567805995944267, −8.89134923261869532280675526610, −7.81813081986057295179665370871, −6.506003273619099815713832674135, −4.90235056441160468127102464266, −4.16597755152993467999575722857, −3.25473475771748695639849276649, −1.09843616657425858975153326305, 1.310484320978586333966965043679, 2.86920337547400483726353235593, 3.626058344224692650726300806842, 5.76893990144516828740120944477, 6.4044436324681813251490851858, 7.681782196092668270411433632731, 8.421378879348906988938767032831, 9.63054905046504533074054312609, 11.22380340698745084599611651817, 11.850434611997915680486336124778, 12.79112858553858022562442616567, 14.008860612085501625240658428928, 14.775221464679912456366221546431, 15.74471931336287452459482089727, 16.994701257984355386176508909359, 18.25783122925138703964434927705, 18.81535837288927256475607671905, 19.41424101219689907857081130071, 20.57697749209444874071137338058, 21.819617705696866716738890622538, 22.857689646780646217382051433989, 23.36944194634275436758358183001, 24.65543022636577132316270837440, 25.27966605190780700142889292949, 26.06408768106172830270934556552

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