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Properties

Label 1-113-113.9-r0-0-0
Degree $1$
Conductor $113$
Sign $-0.497 + 0.867i$
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 + (0.993 + 0.111i)3-s + (−0.900 + 0.433i)4-s + (−0.993 + 0.111i)5-s + (0.111 + 0.993i)6-s + (0.623 + 0.781i)7-s + (−0.623 − 0.781i)8-s + (0.974 + 0.222i)9-s + (−0.330 − 0.943i)10-s + (−0.433 + 0.900i)11-s + (−0.943 + 0.330i)12-s + (−0.781 + 0.623i)13-s + (−0.623 + 0.781i)14-s − 15-s + (0.623 − 0.781i)16-s + (0.846 + 0.532i)17-s + ⋯
L(s)  = 1  + (0.222 + 0.974i)2-s + (0.993 + 0.111i)3-s + (−0.900 + 0.433i)4-s + (−0.993 + 0.111i)5-s + (0.111 + 0.993i)6-s + (0.623 + 0.781i)7-s + (−0.623 − 0.781i)8-s + (0.974 + 0.222i)9-s + (−0.330 − 0.943i)10-s + (−0.433 + 0.900i)11-s + (−0.943 + 0.330i)12-s + (−0.781 + 0.623i)13-s + (−0.623 + 0.781i)14-s − 15-s + (0.623 − 0.781i)16-s + (0.846 + 0.532i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.497 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.497 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(113\)
Sign: $-0.497 + 0.867i$
Analytic conductor: \(0.524769\)
Root analytic conductor: \(0.524769\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{113} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 113,\ (0:\ ),\ -0.497 + 0.867i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6282057780 + 1.084533827i\)
\(L(\frac12)\) \(\approx\) \(0.6282057780 + 1.084533827i\)
\(L(1)\) \(\approx\) \(0.9536201240 + 0.8070104969i\)
\(L(1)\) \(\approx\) \(0.9536201240 + 0.8070104969i\)

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 \( 1 + (0.993 + 0.111i)T \)
5 \( 1 + (-0.993 + 0.111i)T \)
7 \( 1 + (0.623 + 0.781i)T \)
11 \( 1 + (-0.433 + 0.900i)T \)
13 \( 1 + (-0.781 + 0.623i)T \)
17 \( 1 + (0.846 + 0.532i)T \)
19 \( 1 + (0.111 - 0.993i)T \)
23 \( 1 + (-0.111 - 0.993i)T \)
29 \( 1 + (-0.846 - 0.532i)T \)
31 \( 1 + (0.781 - 0.623i)T \)
37 \( 1 + (0.330 + 0.943i)T \)
41 \( 1 + (-0.433 - 0.900i)T \)
43 \( 1 + (0.532 - 0.846i)T \)
47 \( 1 + (-0.943 - 0.330i)T \)
53 \( 1 + (0.900 - 0.433i)T \)
59 \( 1 + (-0.111 + 0.993i)T \)
61 \( 1 + (0.433 - 0.900i)T \)
67 \( 1 + (0.943 - 0.330i)T \)
71 \( 1 + (-0.707 + 0.707i)T \)
73 \( 1 + (0.707 + 0.707i)T \)
79 \( 1 + (-0.330 + 0.943i)T \)
83 \( 1 + (-0.222 - 0.974i)T \)
89 \( 1 + (-0.532 - 0.846i)T \)
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

−29.54452728839392261792362673140, −27.73881262515869591561522865065, −27.09382778156860129690796973363, −26.47625883492565153069243804361, −24.73110879464181308719667637759, −23.793042545760841701359814741687, −22.95239478811036439547462917826, −21.43611766803441063813039495512, −20.6321003004700336009492304472, −19.8055663864217860024285189313, −19.06984457800932980827957341219, −18.018687416591859006759968983127, −16.349159948799277360068639410377, −14.89264956120636174224828177090, −14.14662694092014589367012823684, −13.05633745831804929274408433381, −11.95962325312073916754391768575, −10.792225890842469153150994172311, −9.682495657609581933935214767740, −8.212504990021077603922191520799, −7.61464677586825648712215038646, −5.141274900796672860480804846632, −3.842429935209218572155137948698, −3.004867385372119946071917822084, −1.18348570463836196695917500476, 2.52807106521151474318575444811, 4.09924599186831232217522595211, 5.001787317122234240919209835037, 6.96913163318234668112775542233, 7.85999940380243511607651753957, 8.68243154834331215662671972046, 9.905241979756074207301943580091, 11.89515032180215444015109828976, 12.864248497179925039520542979878, 14.327557868841575280299231196633, 15.07094563083790958645366762107, 15.59336987731730476191978532827, 16.97971749257734850210070177426, 18.427273168081107430062692121514, 19.12072921576462152674423129181, 20.50159879230916531333859212793, 21.58032707445312538923378983947, 22.65544220340665123284237321433, 24.020290644729437319315205697576, 24.43792006127996956944448802394, 25.74926424055727883999245743323, 26.387797123549377643366581540564, 27.42627539818997014780280549539, 28.19607393117435005360987769751, 30.49784642381664004379503231511

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