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Properties

Label 1-232-232.227-r1-0-0
Degree $1$
Conductor $232$
Sign $0.0872 - 0.996i$
Analytic cond. $24.9318$
Root an. cond. $24.9318$
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  + (−0.900 + 0.433i)3-s + (0.222 − 0.974i)5-s + (0.900 − 0.433i)7-s + (0.623 − 0.781i)9-s + (0.623 + 0.781i)11-s + (−0.623 − 0.781i)13-s + (0.222 + 0.974i)15-s + 17-s + (−0.900 − 0.433i)19-s + (−0.623 + 0.781i)21-s + (0.222 + 0.974i)23-s + (−0.900 − 0.433i)25-s + (−0.222 + 0.974i)27-s + (0.222 − 0.974i)31-s + (−0.900 − 0.433i)33-s + ⋯
L(s)  = 1  + (−0.900 + 0.433i)3-s + (0.222 − 0.974i)5-s + (0.900 − 0.433i)7-s + (0.623 − 0.781i)9-s + (0.623 + 0.781i)11-s + (−0.623 − 0.781i)13-s + (0.222 + 0.974i)15-s + 17-s + (−0.900 − 0.433i)19-s + (−0.623 + 0.781i)21-s + (0.222 + 0.974i)23-s + (−0.900 − 0.433i)25-s + (−0.222 + 0.974i)27-s + (0.222 − 0.974i)31-s + (−0.900 − 0.433i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0872 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0872 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(232\)    =    \(2^{3} \cdot 29\)
Sign: $0.0872 - 0.996i$
Analytic conductor: \(24.9318\)
Root analytic conductor: \(24.9318\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{232} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 232,\ (1:\ ),\ 0.0872 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.027736948 - 0.9416653947i\)
\(L(\frac12)\) \(\approx\) \(1.027736948 - 0.9416653947i\)
\(L(1)\) \(\approx\) \(0.9253416780 - 0.2080229397i\)
\(L(1)\) \(\approx\) \(0.9253416780 - 0.2080229397i\)

Euler product

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

−26.48549707352792640918281249910, −25.11438453788149110275858849203, −24.46929905597781067588114042769, −23.473130805206593485730742121651, −22.64212714365894882845724254979, −21.63989176386489865501775203571, −21.22160774934049900116795528863, −19.24282304358090784401686604440, −18.84825860427199115846909619273, −17.84053236103342271655486076267, −17.06050312830140844949173565011, −16.111540746291938940461662174718, −14.51007502424515285181248726944, −14.27356664570771373316583202209, −12.66608092115370360891813824070, −11.71790598022315983314573051461, −11.028614109712383697990010196166, −10.07389716755121763661111621953, −8.56629726499149886306631654619, −7.36089049457468032998012137133, −6.41031418668674398597994363952, −5.53548581422519171236010901368, −4.24143955458186300422227156509, −2.53491832410367597444250808900, −1.32715245241937333895887365998, 0.561852334178081722649755589630, 1.72376430532616193595104492881, 3.92506015200188382940954123712, 4.87135214541369457488234166316, 5.56290684909399764094521532516, 7.02234017673602232625960909680, 8.17722485221350926033300845694, 9.51603986637870002017504689877, 10.261145578978118707984025369473, 11.50734877064253687165445238496, 12.25865804936801452429845728565, 13.22001005951017882120935038293, 14.65623781405471657523172575807, 15.45307846145825107149739139234, 16.8008052906563553079574414638, 17.23156250370402195711119097845, 17.8968037733226929435008054091, 19.47726464205840232783862656958, 20.56617661202720694014205017041, 21.12912708583724105912546132901, 22.1478043444644846022623584033, 23.19260984664047687892499993769, 23.91471505571425206474259771920, 24.79117315633548085978681287664, 25.81116225504502957531757740385

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