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Properties

Label 1-113-113.50-r0-0-0
Degree $1$
Conductor $113$
Sign $0.987 - 0.157i$
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 + (−0.846 − 0.532i)3-s + (0.623 + 0.781i)4-s + (0.846 − 0.532i)5-s + (−0.532 − 0.846i)6-s + (−0.222 − 0.974i)7-s + (0.222 + 0.974i)8-s + (0.433 + 0.900i)9-s + (0.993 − 0.111i)10-s + (−0.781 − 0.623i)11-s + (−0.111 − 0.993i)12-s + (0.974 − 0.222i)13-s + (0.222 − 0.974i)14-s − 15-s + (−0.222 + 0.974i)16-s + (0.943 − 0.330i)17-s + ⋯
L(s)  = 1  + (0.900 + 0.433i)2-s + (−0.846 − 0.532i)3-s + (0.623 + 0.781i)4-s + (0.846 − 0.532i)5-s + (−0.532 − 0.846i)6-s + (−0.222 − 0.974i)7-s + (0.222 + 0.974i)8-s + (0.433 + 0.900i)9-s + (0.993 − 0.111i)10-s + (−0.781 − 0.623i)11-s + (−0.111 − 0.993i)12-s + (0.974 − 0.222i)13-s + (0.222 − 0.974i)14-s − 15-s + (−0.222 + 0.974i)16-s + (0.943 − 0.330i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(113\)
Sign: $0.987 - 0.157i$
Analytic conductor: \(0.524769\)
Root analytic conductor: \(0.524769\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{113} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 113,\ (0:\ ),\ 0.987 - 0.157i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.479035006 - 0.1175223615i\)
\(L(\frac12)\) \(\approx\) \(1.479035006 - 0.1175223615i\)
\(L(1)\) \(\approx\) \(1.444135177 + 0.02009703257i\)
\(L(1)\) \(\approx\) \(1.444135177 + 0.02009703257i\)

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

−29.36517508817686320722688685880, −28.36528513304234510278000160974, −28.05138712761911470587963774076, −26.14788970314307103764185783048, −25.318386950238768693210006839688, −23.9277947311262812664027293449, −22.95796928251021479178212528552, −22.192423299700012313995946647289, −21.29508166443725374671221535791, −20.75839693232100177648276857647, −18.83647171038274299175807235483, −18.15712461026876958476172607338, −16.67274256052644161756351984107, −15.44335517077756317358706682193, −14.79416467268607119529427248923, −13.24897116460739545278116259862, −12.39617295281868692819124347948, −11.12839699268784232911111949865, −10.34589821725579863765209970043, −9.257269037057166222722057879206, −6.77893284618080555484874878653, −5.83106627987470704654654504547, −5.02977581576993120325609856493, −3.39897012523929083162643275997, −1.99031384178781645899981583226, 1.53140963373179207068919320721, 3.533040665568869636876383018978, 5.19224207053786084741185761323, 5.85501174040913964997171371618, 7.05878836914252825789935122141, 8.230946528005892775341948195399, 10.279705922992249274393013321915, 11.27509066366442507948842724673, 12.761386786505182172848967160074, 13.27133798308755263346768109134, 14.172492859911285923345616244569, 16.043279269193389265020604544379, 16.6570178976809793648627390818, 17.519245071439330044660388902919, 18.77593510388552532643917408173, 20.52165315972081755429772017773, 21.21653620569230719182907229614, 22.39090296921608307272302022012, 23.47494025944009904498339311698, 23.84052964943474393307648003031, 25.162478203274113994536203088458, 25.81350292128154168729039228959, 27.36063780312509498536824108900, 28.78053397741300020227453897559, 29.55902048231721652189025627125

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