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Properties

Label 1-113-113.49-r0-0-0
Degree $1$
Conductor $113$
Sign $0.922 - 0.386i$
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 + (0.623 + 0.781i)3-s + (−0.900 + 0.433i)4-s + (0.623 − 0.781i)5-s + (0.623 − 0.781i)6-s + (0.623 + 0.781i)7-s + (0.623 + 0.781i)8-s + (−0.222 + 0.974i)9-s + (−0.900 − 0.433i)10-s + (−0.900 − 0.433i)11-s + (−0.900 − 0.433i)12-s + (0.623 + 0.781i)13-s + (0.623 − 0.781i)14-s + 15-s + (0.623 − 0.781i)16-s + (−0.222 − 0.974i)17-s + ⋯
L(s)  = 1  + (−0.222 − 0.974i)2-s + (0.623 + 0.781i)3-s + (−0.900 + 0.433i)4-s + (0.623 − 0.781i)5-s + (0.623 − 0.781i)6-s + (0.623 + 0.781i)7-s + (0.623 + 0.781i)8-s + (−0.222 + 0.974i)9-s + (−0.900 − 0.433i)10-s + (−0.900 − 0.433i)11-s + (−0.900 − 0.433i)12-s + (0.623 + 0.781i)13-s + (0.623 − 0.781i)14-s + 15-s + (0.623 − 0.781i)16-s + (−0.222 − 0.974i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(113\)
Sign: $0.922 - 0.386i$
Analytic conductor: \(0.524769\)
Root analytic conductor: \(0.524769\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{113} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 113,\ (0:\ ),\ 0.922 - 0.386i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.146019059 - 0.2301773068i\)
\(L(\frac12)\) \(\approx\) \(1.146019059 - 0.2301773068i\)
\(L(1)\) \(\approx\) \(1.129385983 - 0.2253062363i\)
\(L(1)\) \(\approx\) \(1.129385983 - 0.2253062363i\)

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 \( 1 + (0.623 + 0.781i)T \)
5 \( 1 + (0.623 - 0.781i)T \)
7 \( 1 + (0.623 + 0.781i)T \)
11 \( 1 + (-0.900 - 0.433i)T \)
13 \( 1 + (0.623 + 0.781i)T \)
17 \( 1 + (-0.222 - 0.974i)T \)
19 \( 1 + (0.623 + 0.781i)T \)
23 \( 1 + (0.623 - 0.781i)T \)
29 \( 1 + (-0.222 - 0.974i)T \)
31 \( 1 + (0.623 + 0.781i)T \)
37 \( 1 + (-0.900 - 0.433i)T \)
41 \( 1 + (-0.900 + 0.433i)T \)
43 \( 1 + (-0.222 - 0.974i)T \)
47 \( 1 + (-0.900 + 0.433i)T \)
53 \( 1 + (-0.900 + 0.433i)T \)
59 \( 1 + (0.623 + 0.781i)T \)
61 \( 1 + (-0.900 - 0.433i)T \)
67 \( 1 + (-0.900 - 0.433i)T \)
71 \( 1 + T \)
73 \( 1 + T \)
79 \( 1 + (-0.900 + 0.433i)T \)
83 \( 1 + (-0.222 - 0.974i)T \)
89 \( 1 + (-0.222 + 0.974i)T \)
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

−29.62840162840260844708038736492, −28.33878384362923512116495262952, −26.9469358183399804740573658372, −26.01814912588732849919381012473, −25.62596823707676598541814851466, −24.364746411744606120376996712759, −23.59040587411183785390011393200, −22.6715368055503513253939282188, −21.15832611167033868771321949869, −19.93675553456582839854713558781, −18.69626397638536980660327068231, −17.84271404262276782304178770266, −17.33667377751079706340328128083, −15.45754001020621007472173703722, −14.72269389898181619924707960963, −13.58098332473680713319780621671, −13.120288532010097976417499968927, −10.91010530862367796920441465291, −9.81374387078874905783674314180, −8.36539603342969353123152377551, −7.47447496731973865309414763287, −6.59013764985913513125447776400, −5.2376644887681547526543614958, −3.35272776301203731112872568498, −1.51694243040231516803470717446, 1.796952865094316907640273560794, 2.96456615362612651489070049950, 4.57936016243303669331474492770, 5.40199655112450521296400622743, 8.16323545282552769719837629675, 8.84920581534006468001625002287, 9.764035859795019428854227563546, 10.93652896004304942660131771092, 12.093910208289214604389366385180, 13.48441476934131753955892212147, 14.142985226853079486436105048705, 15.768692371012579449165426941084, 16.76717652662983307508154665066, 18.1512540979494162582876735109, 18.976683085687304044416564378097, 20.47276104566329568625413995909, 20.9767165190416603071408393726, 21.536259407910495063359134122312, 22.77116594744786441739383961556, 24.40561210438059206541647735924, 25.409387286050346131993221016736, 26.553269190888683716791578898730, 27.33585991982636739245432402359, 28.54444074765223726942072538020, 28.81952115141552674961546370580

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