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Properties

Label 1-113-113.88-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: $\chi_{113} (88, \cdot )$
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

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

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