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Properties

Label 1-232-232.123-r1-0-0
Degree $1$
Conductor $232$
Sign $0.999 + 0.00388i$
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  + (0.623 + 0.781i)3-s + (0.900 − 0.433i)5-s + (−0.623 − 0.781i)7-s + (−0.222 + 0.974i)9-s + (−0.222 − 0.974i)11-s + (0.222 + 0.974i)13-s + (0.900 + 0.433i)15-s + 17-s + (0.623 − 0.781i)19-s + (0.222 − 0.974i)21-s + (0.900 + 0.433i)23-s + (0.623 − 0.781i)25-s + (−0.900 + 0.433i)27-s + (0.900 − 0.433i)31-s + (0.623 − 0.781i)33-s + ⋯
L(s)  = 1  + (0.623 + 0.781i)3-s + (0.900 − 0.433i)5-s + (−0.623 − 0.781i)7-s + (−0.222 + 0.974i)9-s + (−0.222 − 0.974i)11-s + (0.222 + 0.974i)13-s + (0.900 + 0.433i)15-s + 17-s + (0.623 − 0.781i)19-s + (0.222 − 0.974i)21-s + (0.900 + 0.433i)23-s + (0.623 − 0.781i)25-s + (−0.900 + 0.433i)27-s + (0.900 − 0.433i)31-s + (0.623 − 0.781i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.00388i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.00388i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(232\)    =    \(2^{3} \cdot 29\)
Sign: $0.999 + 0.00388i$
Analytic conductor: \(24.9318\)
Root analytic conductor: \(24.9318\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{232} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 232,\ (1:\ ),\ 0.999 + 0.00388i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.741065535 + 0.005318318838i\)
\(L(\frac12)\) \(\approx\) \(2.741065535 + 0.005318318838i\)
\(L(1)\) \(\approx\) \(1.546045319 + 0.09332882627i\)
\(L(1)\) \(\approx\) \(1.546045319 + 0.09332882627i\)

Euler product

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

−25.69783929845706309926495308716, −25.25804631092823873223069735845, −24.68144687187927731463457975285, −23.086388134811598164835391499417, −22.63924186996516854571663464416, −21.2594741171191065885714673560, −20.54589807253479785441213714968, −19.42673940943802316657191256717, −18.4082868857240862632306719805, −18.047029953794550744167945330272, −16.82215941543940527108944526583, −15.3172241188175815255499837479, −14.69070645453527300324828596023, −13.5699609906606981371599334923, −12.756194867019571487193677194536, −12.00759097535058989347743470132, −10.24622162991665339257012930372, −9.58109920357612484564039186340, −8.40622572732634401564379375693, −7.28892520049181492713492775659, −6.26892220273163293255893809535, −5.348134884289834169111706325477, −3.245637576071049176337465195, −2.53415447866592927482251832356, −1.231250365410064275793562690364, 0.98930524868811695153940766792, 2.66186911169308940639403667775, 3.71631221104728968259079862596, 4.93486732376143379353132149798, 6.03818463165851644089405021250, 7.42066741178559670251384283092, 8.773122036199980903527119513132, 9.49261607190813769782747497940, 10.33014150365150692473570733602, 11.39427007022148726028158656403, 13.09729533716869767794016673215, 13.71724877034543415290029283623, 14.43739486149422518650400634313, 15.94695606382225434318955422206, 16.489318590313409962482239705672, 17.32815504993639654655262997305, 18.854257126738688411550629765, 19.60863311254727460458410399426, 20.74689726306240412791048485333, 21.28217110814696253601993884389, 22.110346317886596264075482254084, 23.29749394253146327004935076241, 24.39714621012138107902286538436, 25.34462743966978947568420768438, 26.233155306915485341700739192389

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