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Properties

Label 1-113-113.77-r0-0-0
Degree $1$
Conductor $113$
Sign $-0.996 - 0.0829i$
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 + (−0.111 − 0.993i)3-s + (−0.900 − 0.433i)4-s + (0.111 − 0.993i)5-s + (−0.993 − 0.111i)6-s + (0.623 − 0.781i)7-s + (−0.623 + 0.781i)8-s + (−0.974 + 0.222i)9-s + (−0.943 − 0.330i)10-s + (0.433 + 0.900i)11-s + (−0.330 + 0.943i)12-s + (0.781 + 0.623i)13-s + (−0.623 − 0.781i)14-s − 15-s + (0.623 + 0.781i)16-s + (−0.532 − 0.846i)17-s + ⋯
L(s)  = 1  + (0.222 − 0.974i)2-s + (−0.111 − 0.993i)3-s + (−0.900 − 0.433i)4-s + (0.111 − 0.993i)5-s + (−0.993 − 0.111i)6-s + (0.623 − 0.781i)7-s + (−0.623 + 0.781i)8-s + (−0.974 + 0.222i)9-s + (−0.943 − 0.330i)10-s + (0.433 + 0.900i)11-s + (−0.330 + 0.943i)12-s + (0.781 + 0.623i)13-s + (−0.623 − 0.781i)14-s − 15-s + (0.623 + 0.781i)16-s + (−0.532 − 0.846i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0829i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(113\)
Sign: $-0.996 - 0.0829i$
Analytic conductor: \(0.524769\)
Root analytic conductor: \(0.524769\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{113} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 113,\ (0:\ ),\ -0.996 - 0.0829i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04309342781 - 1.037790878i\)
\(L(\frac12)\) \(\approx\) \(0.04309342781 - 1.037790878i\)
\(L(1)\) \(\approx\) \(0.5554757202 - 0.9019761041i\)
\(L(1)\) \(\approx\) \(0.5554757202 - 0.9019761041i\)

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

−30.2254913144518317454144040204, −28.50917559830957793739927774003, −27.382134104326303396216451830337, −26.864619654631946296972535232891, −25.75898680266361745084037498854, −25.01746776305246252932692871161, −23.60691811462736228024566478044, −22.67047005650861826453242475433, −21.69465206610465899319018097111, −21.27455260533050581263867868785, −19.29557295075458785243412620843, −18.104067054251251298143291992902, −17.24861815537324707542297616137, −16.024685950729129385853887447235, −15.00616413350163240518943650128, −14.62052254319856910374068172184, −13.24122287534390092562851466964, −11.47240738653339245437168121991, −10.538788848446252447672178154427, −8.98849778068295516263613578061, −8.23181020062287849452658768062, −6.38621394531389716767908893221, −5.66827303412748795978322505733, −4.2093102832301267759162047370, −3.007845693050336997273633431098, 1.07808220784879790166172610007, 2.05820045809687068348661240809, 4.11066631055033819335944300164, 5.15206708086660127262501874330, 6.8064724479854845312864614371, 8.33147756697618117188220510839, 9.3017373063831669905812545779, 10.942591110679184272047625648588, 11.82821099410227198205130486670, 12.906776990730696098776774020506, 13.58511197328548039311635055144, 14.67232612570754196501260372984, 16.757063059996235127931714127534, 17.592569050397116681546323615912, 18.55013815349532273938330447848, 19.87649577402543235831930568096, 20.36302377510801170095136735911, 21.417202660720430647748040742539, 23.01799173944834163328421012113, 23.540770211558643284161069166090, 24.52411900547895605263089472827, 25.696611221891512402873032276127, 27.29524947604766540560571102475, 28.10704589080300004382081974338, 29.05314404946790779778002160998

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