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Properties

Label 1-113-113.32-r0-0-0
Degree $1$
Conductor $113$
Sign $0.675 - 0.736i$
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  + (−0.900 + 0.433i)2-s + (−0.974 − 0.222i)3-s + (0.623 − 0.781i)4-s + (−0.974 + 0.222i)5-s + (0.974 − 0.222i)6-s + (−0.222 + 0.974i)7-s + (−0.222 + 0.974i)8-s + (0.900 + 0.433i)9-s + (0.781 − 0.623i)10-s + (−0.623 − 0.781i)11-s + (−0.781 + 0.623i)12-s + (0.222 − 0.974i)13-s + (−0.222 − 0.974i)14-s + 15-s + (−0.222 − 0.974i)16-s + (−0.433 − 0.900i)17-s + ⋯
L(s)  = 1  + (−0.900 + 0.433i)2-s + (−0.974 − 0.222i)3-s + (0.623 − 0.781i)4-s + (−0.974 + 0.222i)5-s + (0.974 − 0.222i)6-s + (−0.222 + 0.974i)7-s + (−0.222 + 0.974i)8-s + (0.900 + 0.433i)9-s + (0.781 − 0.623i)10-s + (−0.623 − 0.781i)11-s + (−0.781 + 0.623i)12-s + (0.222 − 0.974i)13-s + (−0.222 − 0.974i)14-s + 15-s + (−0.222 − 0.974i)16-s + (−0.433 − 0.900i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.675 - 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.675 - 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(113\)
Sign: $0.675 - 0.736i$
Analytic conductor: \(0.524769\)
Root analytic conductor: \(0.524769\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{113} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 113,\ (0:\ ),\ 0.675 - 0.736i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3022436065 - 0.1329065737i\)
\(L(\frac12)\) \(\approx\) \(0.3022436065 - 0.1329065737i\)
\(L(1)\) \(\approx\) \(0.4303997469 + 0.002723265735i\)
\(L(1)\) \(\approx\) \(0.4303997469 + 0.002723265735i\)

Euler product

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

−28.932955094815012191805561076029, −28.661092394130916666361677264399, −27.46543336758437219760371474612, −26.76361777158076072433868678668, −25.90870869064162153626594923169, −24.130657954882078233455527171282, −23.449074134411566675095300685525, −22.32412175617748517836995505941, −21.03781161936140757585818232893, −20.13517823874862158031729883896, −19.11247404757848400298452313465, −18.00298233282695255048291660295, −16.96783501308285393221406195197, −16.22600357910638234157284272983, −15.307586388688464303131403423102, −13.09909131052129630904358691652, −12.10990857536681048579194563108, −11.10822475079864640831829729985, −10.34906121237621720442565990845, −9.061813077507121348538020846523, −7.484194021848572010384054755844, −6.80741971669441690520349606994, −4.68869874035235033528750318053, −3.583326688251466904499409201742, −1.25134712341633914917737029856, 0.55707703020588658028638297311, 2.80904545625822780577414448693, 5.135296385250652831267787655422, 6.08663939047617456104752348868, 7.38443164911854939095293093521, 8.31103678049257014300193014911, 9.791240738803188331498738912915, 11.14949790742663824734131234376, 11.64359781222628677157225104118, 13.099542829752945853438325303071, 15.07850616093973727393149716059, 15.835015725310596163375304877027, 16.535309844433008656542624981928, 18.08170944131423426224151758558, 18.50888449188568678627365504104, 19.474075778979570169617846524677, 20.86213601436264328299417547492, 22.449042397022903243082684171857, 23.14014462939028173527098551411, 24.37010501798592585760341583952, 24.91681188348203661810065914754, 26.51329084352762349815415385060, 27.25526252463074399468252012383, 28.14743807808525589975037414283, 28.90450958308422437000613630363

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