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Properties

Label 1-113-113.22-r0-0-0
Degree $1$
Conductor $113$
Sign $0.956 - 0.292i$
Analytic cond. $0.524769$
Root an. cond. $0.524769$
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)2-s + (0.111 − 0.993i)3-s + (−0.900 + 0.433i)4-s + (−0.111 − 0.993i)5-s + (0.993 − 0.111i)6-s + (0.623 + 0.781i)7-s + (−0.623 − 0.781i)8-s + (−0.974 − 0.222i)9-s + (0.943 − 0.330i)10-s + (0.433 − 0.900i)11-s + (0.330 + 0.943i)12-s + (0.781 − 0.623i)13-s + (−0.623 + 0.781i)14-s − 15-s + (0.623 − 0.781i)16-s + (0.532 − 0.846i)17-s + ⋯
L(s)  = 1  + (0.222 + 0.974i)2-s + (0.111 − 0.993i)3-s + (−0.900 + 0.433i)4-s + (−0.111 − 0.993i)5-s + (0.993 − 0.111i)6-s + (0.623 + 0.781i)7-s + (−0.623 − 0.781i)8-s + (−0.974 − 0.222i)9-s + (0.943 − 0.330i)10-s + (0.433 − 0.900i)11-s + (0.330 + 0.943i)12-s + (0.781 − 0.623i)13-s + (−0.623 + 0.781i)14-s − 15-s + (0.623 − 0.781i)16-s + (0.532 − 0.846i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(113\)
Sign: $0.956 - 0.292i$
Analytic conductor: \(0.524769\)
Root analytic conductor: \(0.524769\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{113} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 113,\ (0:\ ),\ 0.956 - 0.292i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.097486406 - 0.1640547591i\)
\(L(\frac12)\) \(\approx\) \(1.097486406 - 0.1640547591i\)
\(L(1)\) \(\approx\) \(1.117492039 + 0.006025693460i\)
\(L(1)\) \(\approx\) \(1.117492039 + 0.006025693460i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad113 \( 1 \)
good2 \( 1 + (0.222 + 0.974i)T \)
3 \( 1 + (0.111 - 0.993i)T \)
5 \( 1 + (-0.111 - 0.993i)T \)
7 \( 1 + (0.623 + 0.781i)T \)
11 \( 1 + (0.433 - 0.900i)T \)
13 \( 1 + (0.781 - 0.623i)T \)
17 \( 1 + (0.532 - 0.846i)T \)
19 \( 1 + (0.993 + 0.111i)T \)
23 \( 1 + (-0.993 + 0.111i)T \)
29 \( 1 + (-0.532 + 0.846i)T \)
31 \( 1 + (-0.781 + 0.623i)T \)
37 \( 1 + (-0.943 + 0.330i)T \)
41 \( 1 + (0.433 + 0.900i)T \)
43 \( 1 + (0.846 + 0.532i)T \)
47 \( 1 + (0.330 - 0.943i)T \)
53 \( 1 + (0.900 - 0.433i)T \)
59 \( 1 + (-0.993 - 0.111i)T \)
61 \( 1 + (-0.433 + 0.900i)T \)
67 \( 1 + (-0.330 - 0.943i)T \)
71 \( 1 + (0.707 + 0.707i)T \)
73 \( 1 + (-0.707 + 0.707i)T \)
79 \( 1 + (0.943 + 0.330i)T \)
83 \( 1 + (-0.222 - 0.974i)T \)
89 \( 1 + (-0.846 + 0.532i)T \)
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

−29.53597417143572749272694654728, −28.17192134783889536548112741202, −27.60292813866027203022861315604, −26.45529119069524166039913723034, −25.97125264493142466952273382994, −23.87346490886676549993383500227, −22.8535245494850671270837885506, −22.19924137939385500639690156673, −21.07874785937307671109217910535, −20.36144516057521584330854528187, −19.33599189197350070025564090584, −18.075149021959762372092545881284, −17.086158116446170190771368782144, −15.44686041021568393253795702088, −14.38452557437364702760042120877, −13.85944910450990618896192970945, −11.96016776398699796081195086266, −11.00671558488900886689061402757, −10.28355927253519224408937093422, −9.26338008773920663676971506024, −7.698430957713897475766633965726, −5.82655798844304481835367480687, −4.21746831297709136740997154176, −3.65573404121684917138210946646, −1.97291837367395322292740928224, 1.17313658565236465054047579084, 3.384427423238084902376130910538, 5.27536124838999608642581166112, 5.90619045394366420724681111419, 7.52867583199911276640136125642, 8.41396238328565561772274315729, 9.1359454334637646969436737335, 11.62475053777344726696110089054, 12.454856822468528867507982044883, 13.57449342073248799310546951781, 14.35814335174875447784713373724, 15.806567324993024483004304097115, 16.66420869408085966631280448068, 17.95855606540093660472333454364, 18.4860949693640201401759919368, 19.9657010326086208193283176061, 21.164068449077717720980574861362, 22.44997850639745582661736018525, 23.59178767674363220396953940786, 24.45876376665432260933547978754, 24.85387165005411082659814176104, 25.83792992182471961524203193026, 27.37444124504363472496191649794, 28.06920406346669071865611443160, 29.4270107787638932888144101569

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