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Properties

Label 1-113-113.109-r0-0-0
Degree $1$
Conductor $113$
Sign $-0.828 - 0.560i$
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 + (−0.222 − 0.974i)3-s + (0.623 + 0.781i)4-s + (−0.222 + 0.974i)5-s + (−0.222 + 0.974i)6-s + (−0.222 − 0.974i)7-s + (−0.222 − 0.974i)8-s + (−0.900 + 0.433i)9-s + (0.623 − 0.781i)10-s + (0.623 − 0.781i)11-s + (0.623 − 0.781i)12-s + (−0.222 − 0.974i)13-s + (−0.222 + 0.974i)14-s + 15-s + (−0.222 + 0.974i)16-s + (−0.900 − 0.433i)17-s + ⋯
L(s)  = 1  + (−0.900 − 0.433i)2-s + (−0.222 − 0.974i)3-s + (0.623 + 0.781i)4-s + (−0.222 + 0.974i)5-s + (−0.222 + 0.974i)6-s + (−0.222 − 0.974i)7-s + (−0.222 − 0.974i)8-s + (−0.900 + 0.433i)9-s + (0.623 − 0.781i)10-s + (0.623 − 0.781i)11-s + (0.623 − 0.781i)12-s + (−0.222 − 0.974i)13-s + (−0.222 + 0.974i)14-s + 15-s + (−0.222 + 0.974i)16-s + (−0.900 − 0.433i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.828 - 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.828 - 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(113\)
Sign: $-0.828 - 0.560i$
Analytic conductor: \(0.524769\)
Root analytic conductor: \(0.524769\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{113} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 113,\ (0:\ ),\ -0.828 - 0.560i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1326591108 - 0.4329357822i\)
\(L(\frac12)\) \(\approx\) \(0.1326591108 - 0.4329357822i\)
\(L(1)\) \(\approx\) \(0.4556900067 - 0.3216846877i\)
\(L(1)\) \(\approx\) \(0.4556900067 - 0.3216846877i\)

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

−29.040265079082653872047071864308, −28.47181110415941158234375372931, −27.78958019760392223596803605398, −26.90466155469127135683548653002, −25.787452843187659266232671211511, −24.89514653104821305186851313746, −23.90158666085945808095423363425, −22.61974512218611979229465935542, −21.39693966671821534440437030790, −20.35724058834380738171286396658, −19.51897735816811622483357416189, −18.174231988242020936512395259635, −16.93286401061590912714458619253, −16.37999951084144894907070865757, −15.34090086364514219530229442248, −14.53193823215228604552838797244, −12.36872122166849466741672921366, −11.49014356676652301442166786734, −10.02995858330946650333712204179, −9.08709761251658641791444864790, −8.50450365330970674640604197315, −6.6215773725323747646016456085, −5.415340755700487482162978847074, −4.22639783596007554004695998454, −1.9750906888416201875897805823, 0.58302597026991703139198095204, 2.3982561668733577977139519066, 3.61962917863223171788366425174, 6.22480145455169578012092252417, 7.18046156354060191955050319955, 7.94486244790975935464080342851, 9.5110602536910804899796912349, 11.036391765999446718718297591067, 11.30544231800359805724074597987, 12.91317676599940211659643844054, 13.84319254695120825607247768264, 15.43760098248130341569359576257, 16.89020510580883477511399085126, 17.6500711525904174213092367532, 18.569036049573838232048146503491, 19.619251561990300752516034267658, 20.04544045742185425082298700933, 21.893302042241496308822755200286, 22.71584552626382862610972645485, 23.9786225837276612475505179905, 25.04338414406204852412268261072, 26.110986727332821115699856573234, 26.87926948805376657326249566669, 27.922831450071841349407404859283, 29.25391327862860631158468418383

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