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Properties

Label 1-113-113.64-r0-0-0
Degree $1$
Conductor $113$
Sign $0.968 + 0.247i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(113\)
Sign: $0.968 + 0.247i$
Analytic conductor: \(0.524769\)
Root analytic conductor: \(0.524769\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{113} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 113,\ (0:\ ),\ 0.968 + 0.247i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5101833271 + 0.06425107790i\)
\(L(\frac12)\) \(\approx\) \(0.5101833271 + 0.06425107790i\)
\(L(1)\) \(\approx\) \(0.6040773311 - 0.1484669171i\)
\(L(1)\) \(\approx\) \(0.6040773311 - 0.1484669171i\)

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

−28.856085164898033601957953621172, −27.953245779717720355792640405896, −27.27172669417520494227082605621, −26.52402789140678239347673602148, −25.26461969155966790419035537983, −24.02087613838893608312060251865, −23.27796987370604159322580188094, −22.67014722860524989161101086794, −20.91365632003452307150716002165, −20.33669399919777836654004330679, −18.55597932632943472025290112999, −17.51265519281006265577928701288, −16.654257067774551127083112233181, −15.81789519308857650773190790783, −15.01902003349489031901827669252, −13.611692826755421608757803271215, −12.2915113865245868668851189721, −10.794520430175343623411487529866, −9.86668342789835872410138871970, −8.41212438679860734891700653349, −7.57350678080682693288684582806, −5.892108631191764175736873713876, −4.80356096795557230529059717707, −4.02872974661498453122376795697, −0.611910591804225311388793995074, 1.69765266620610753719563248528, 2.98284348311308295013408906591, 4.69744463390995430607118594718, 6.17195493479657481984635121956, 7.7695110178318869242987824459, 8.6008307076573533503106841877, 10.54706105185997845936924901392, 11.25668996930937671794206309737, 12.0491716185152337126336760659, 13.18526225827356994076952738661, 14.34608671245380886928422131563, 15.85501234065080337107362382225, 17.36897575517635820568344644033, 18.36641594652495820384711608570, 18.82012575380104635520995591943, 19.796924564645826135125609396065, 21.50266672085194320551418917490, 21.90815339924964622504123864336, 23.56211500430522833404191924843, 23.63641173866907261395379025097, 25.55024421042967222477888893497, 26.54318244713965128535572120065, 27.809138852988635727699107224120, 28.358456911260555058798911897272, 29.42497666430506059425539305507

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