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Properties

Label 1-113-113.100-r0-0-0
Degree $1$
Conductor $113$
Sign $-0.751 - 0.659i$
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 + (−0.330 − 0.943i)3-s + (−0.222 − 0.974i)4-s + (0.330 − 0.943i)5-s + (0.943 + 0.330i)6-s + (−0.900 − 0.433i)7-s + (0.900 + 0.433i)8-s + (−0.781 + 0.623i)9-s + (0.532 + 0.846i)10-s + (−0.974 − 0.222i)11-s + (−0.846 + 0.532i)12-s + (−0.433 + 0.900i)13-s + (0.900 − 0.433i)14-s − 15-s + (−0.900 + 0.433i)16-s + (−0.993 + 0.111i)17-s + ⋯
L(s)  = 1  + (−0.623 + 0.781i)2-s + (−0.330 − 0.943i)3-s + (−0.222 − 0.974i)4-s + (0.330 − 0.943i)5-s + (0.943 + 0.330i)6-s + (−0.900 − 0.433i)7-s + (0.900 + 0.433i)8-s + (−0.781 + 0.623i)9-s + (0.532 + 0.846i)10-s + (−0.974 − 0.222i)11-s + (−0.846 + 0.532i)12-s + (−0.433 + 0.900i)13-s + (0.900 − 0.433i)14-s − 15-s + (−0.900 + 0.433i)16-s + (−0.993 + 0.111i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.751 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.751 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(113\)
Sign: $-0.751 - 0.659i$
Analytic conductor: \(0.524769\)
Root analytic conductor: \(0.524769\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{113} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 113,\ (0:\ ),\ -0.751 - 0.659i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1198745445 - 0.3185980874i\)
\(L(\frac12)\) \(\approx\) \(0.1198745445 - 0.3185980874i\)
\(L(1)\) \(\approx\) \(0.4847767321 - 0.1648484067i\)
\(L(1)\) \(\approx\) \(0.4847767321 - 0.1648484067i\)

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

−29.17638038944591895570499146658, −28.98312579605002562107272705976, −27.7486514402156506718157772591, −26.70020473789275943909385890089, −26.13946588245033991180024648025, −25.20176392969274830743388939316, −23.004969566368715001234684690594, −22.27958208435829124462158915198, −21.66468688315246326109397051216, −20.468991283597460729247176094617, −19.50746421102256018772822846578, −18.15468232810773712178107087162, −17.63474156130246699326468068372, −16.08808257326222114978565953479, −15.41192946208901844987405356454, −13.76438872653153011000511303911, −12.44130321182344325502823938784, −11.2738519588939764789834051756, −10.06996982881667267318873551669, −9.87368093977100414984960760441, −8.275128193666022343110257748989, −6.677354614004525887281880054466, −5.155466776414635689306897771309, −3.38635285140839936519379948784, −2.622005134230389225016760273928, 0.39294154494863636215823262169, 2.0903882185246314129381545262, 4.7670817855748935920689119254, 5.97059073589228201705005530738, 6.95110698816818717295797251363, 8.09789032996142407789581716269, 9.223186318716256146314607383396, 10.40766214349040327944744527759, 11.988487414427533042337894661620, 13.35411510613456384945531168420, 13.814322403724162388714360721129, 15.779757303481586226767829579731, 16.528455984622549008326932622, 17.43200737650659766481145015117, 18.38715093631060841298938456406, 19.47444479409129313443377810853, 20.245869720307041829021585076846, 22.08305547912523246907978511510, 23.32032862129071582806751171902, 24.101536704268350622240958029051, 24.72604124334560739653179448349, 25.91314947940465500744465284712, 26.6257544289741631748333777545, 28.36167372723832943800080489716, 28.74594601872304859118155105727

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