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Properties

Label 1-113-113.8-r0-0-0
Degree $1$
Conductor $113$
Sign $-0.609 - 0.792i$
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 + (−0.433 + 0.900i)3-s + (−0.222 − 0.974i)4-s + (−0.433 − 0.900i)5-s + (0.433 + 0.900i)6-s + (−0.900 − 0.433i)7-s + (−0.900 − 0.433i)8-s + (−0.623 − 0.781i)9-s + (−0.974 − 0.222i)10-s + (0.222 − 0.974i)11-s + (0.974 + 0.222i)12-s + (0.900 + 0.433i)13-s + (−0.900 + 0.433i)14-s + 15-s + (−0.900 + 0.433i)16-s + (−0.781 − 0.623i)17-s + ⋯
L(s)  = 1  + (0.623 − 0.781i)2-s + (−0.433 + 0.900i)3-s + (−0.222 − 0.974i)4-s + (−0.433 − 0.900i)5-s + (0.433 + 0.900i)6-s + (−0.900 − 0.433i)7-s + (−0.900 − 0.433i)8-s + (−0.623 − 0.781i)9-s + (−0.974 − 0.222i)10-s + (0.222 − 0.974i)11-s + (0.974 + 0.222i)12-s + (0.900 + 0.433i)13-s + (−0.900 + 0.433i)14-s + 15-s + (−0.900 + 0.433i)16-s + (−0.781 − 0.623i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.609 - 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.609 - 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(113\)
Sign: $-0.609 - 0.792i$
Analytic conductor: \(0.524769\)
Root analytic conductor: \(0.524769\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{113} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 113,\ (0:\ ),\ -0.609 - 0.792i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3863923665 - 0.7847339412i\)
\(L(\frac12)\) \(\approx\) \(0.3863923665 - 0.7847339412i\)
\(L(1)\) \(\approx\) \(0.8015082645 - 0.5379572201i\)
\(L(1)\) \(\approx\) \(0.8015082645 - 0.5379572201i\)

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

−30.069141391581280089028057403993, −28.8950856053178387227107201480, −27.688800127622992671575444394509, −26.19441469211718564956808870186, −25.561071962041041965672765259869, −24.609008335340053904933544649973, −23.43918270099477342037243503917, −22.54166151914897775274498059544, −22.382631194169179754745743747779, −20.47321000372568641104831818775, −19.03564733995267718497548485702, −18.233451755859857575685679576028, −17.21403475367053823467322632547, −15.92896503038531948384245807863, −15.05022136276386046923227311120, −13.84088629179865728095244991164, −12.73398908810732478071704702626, −12.02162565514036480733147087857, −10.62143209231445152232391706870, −8.687161124419246924426968765908, −7.40439832366536280792409635462, −6.58728415933927400963008364436, −5.734832915577868578832130667712, −3.89796850110123038189937623140, −2.52571786150138155501272079351, 0.74322340206577396598281787392, 3.25832197086638288539019909496, 4.12510595458511090076880329708, 5.27571959639230853589539463909, 6.47615739027076631842892638564, 8.89396910530052804501805341223, 9.57806647617887596970192265490, 11.0678478582828469494525521780, 11.623613454877862948424599513528, 13.059175573676996228057256470511, 13.871696582319900536984034819185, 15.66239466932258512544492830371, 16.0286593673998236797797000425, 17.374282612295980841712339999411, 19.06263108884807459530210902232, 19.97152673365799462854262804811, 20.83965920734837945859960434769, 21.75246298216269711482626950658, 22.77659089994661807059074659207, 23.52796391395280804580146810796, 24.571946158159895152475985649909, 26.31966731877955526710290939610, 27.23049134921038240157832730316, 28.30072894374397021914558850297, 28.82939875860845695551929736165

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