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Properties

Label 1-113-113.51-r0-0-0
Degree $1$
Conductor $113$
Sign $0.825 + 0.564i$
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  + (−0.623 − 0.781i)2-s + (0.943 + 0.330i)3-s + (−0.222 + 0.974i)4-s + (−0.943 + 0.330i)5-s + (−0.330 − 0.943i)6-s + (−0.900 + 0.433i)7-s + (0.900 − 0.433i)8-s + (0.781 + 0.623i)9-s + (0.846 + 0.532i)10-s + (0.974 − 0.222i)11-s + (−0.532 + 0.846i)12-s + (0.433 + 0.900i)13-s + (0.900 + 0.433i)14-s − 15-s + (−0.900 − 0.433i)16-s + (−0.111 + 0.993i)17-s + ⋯
L(s)  = 1  + (−0.623 − 0.781i)2-s + (0.943 + 0.330i)3-s + (−0.222 + 0.974i)4-s + (−0.943 + 0.330i)5-s + (−0.330 − 0.943i)6-s + (−0.900 + 0.433i)7-s + (0.900 − 0.433i)8-s + (0.781 + 0.623i)9-s + (0.846 + 0.532i)10-s + (0.974 − 0.222i)11-s + (−0.532 + 0.846i)12-s + (0.433 + 0.900i)13-s + (0.900 + 0.433i)14-s − 15-s + (−0.900 − 0.433i)16-s + (−0.111 + 0.993i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.825 + 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.825 + 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(113\)
Sign: $0.825 + 0.564i$
Analytic conductor: \(0.524769\)
Root analytic conductor: \(0.524769\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{113} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 113,\ (0:\ ),\ 0.825 + 0.564i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7780049518 + 0.2405802023i\)
\(L(\frac12)\) \(\approx\) \(0.7780049518 + 0.2405802023i\)
\(L(1)\) \(\approx\) \(0.8541459569 + 0.04336243475i\)
\(L(1)\) \(\approx\) \(0.8541459569 + 0.04336243475i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad113 \( 1 \)
good2 \( 1 + (-0.623 - 0.781i)T \)
3 \( 1 + (0.943 + 0.330i)T \)
5 \( 1 + (-0.943 + 0.330i)T \)
7 \( 1 + (-0.900 + 0.433i)T \)
11 \( 1 + (0.974 - 0.222i)T \)
13 \( 1 + (0.433 + 0.900i)T \)
17 \( 1 + (-0.111 + 0.993i)T \)
19 \( 1 + (-0.330 + 0.943i)T \)
23 \( 1 + (0.330 + 0.943i)T \)
29 \( 1 + (0.111 - 0.993i)T \)
31 \( 1 + (-0.433 - 0.900i)T \)
37 \( 1 + (-0.846 - 0.532i)T \)
41 \( 1 + (0.974 + 0.222i)T \)
43 \( 1 + (-0.993 - 0.111i)T \)
47 \( 1 + (-0.532 - 0.846i)T \)
53 \( 1 + (0.222 - 0.974i)T \)
59 \( 1 + (0.330 - 0.943i)T \)
61 \( 1 + (-0.974 + 0.222i)T \)
67 \( 1 + (0.532 - 0.846i)T \)
71 \( 1 + (0.707 + 0.707i)T \)
73 \( 1 + (-0.707 + 0.707i)T \)
79 \( 1 + (0.846 - 0.532i)T \)
83 \( 1 + (0.623 + 0.781i)T \)
89 \( 1 + (0.993 - 0.111i)T \)
97 \( 1 + (-0.900 - 0.433i)T \)
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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

−29.21423625563657506380910556782, −27.84128672854417768531864872856, −27.10176123660133082836425364092, −26.146032194666771515086132439714, −25.24597290582257620696477950843, −24.46607602978758335612811431107, −23.37495532083171493148816143975, −22.52281598994420297280388030075, −20.3261870184346286597415633096, −19.84570645926701351480675413111, −19.031193817689008650687132097680, −17.890160572209252257533016499991, −16.49197397232953545563205222486, −15.67162779336892907107956799805, −14.73565825960414567874676012058, −13.564090090114279054341148584794, −12.44624847206604062694810390629, −10.70087698316434271881347517542, −9.29263489574883384338888308371, −8.60386676248664697079844622571, −7.32663972272332399664434999544, −6.649049656493218628421076692581, −4.59914684285028672490426492776, −3.19217764528703314642115167888, −0.951959363717896116072025380581, 1.9472971630498126267019161290, 3.54288484166213287006471958003, 3.94133868834215986868181004602, 6.67681182496648137065581846425, 8.02508773199581625651904558379, 8.94375729695069719090115277708, 9.86999801686166506934307607223, 11.1745588491285471410323336765, 12.24241418433184768809766090686, 13.41525340284475544166184954114, 14.76608693292765804059205690065, 15.91705402564994817758918269779, 16.82597572204337652381012512137, 18.675562771981186274938513031456, 19.26957587146898572901748169440, 19.79357459314348871178171645625, 21.12440374492008298309971496519, 21.9663729764767940051082876242, 23.04184868218404022154314441071, 24.71965388673218274990039992866, 25.849419466512112451491965434017, 26.43770714341329400109795944283, 27.46762925229603489159070744365, 28.18329137703529488096046776813, 29.56287835902926340807291464410

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