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Properties

Label 1-109-109.44-r1-0-0
Degree $1$
Conductor $109$
Sign $-0.139 - 0.990i$
Analytic cond. $11.7136$
Root an. cond. $11.7136$
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  + (0.984 − 0.173i)2-s + (0.597 − 0.802i)3-s + (0.939 − 0.342i)4-s + (−0.686 − 0.727i)5-s + (0.448 − 0.893i)6-s + (0.973 − 0.230i)7-s + (0.866 − 0.5i)8-s + (−0.286 − 0.957i)9-s + (−0.802 − 0.597i)10-s + (−0.802 + 0.597i)11-s + (0.286 − 0.957i)12-s + (0.230 + 0.973i)13-s + (0.918 − 0.396i)14-s + (−0.993 + 0.116i)15-s + (0.766 − 0.642i)16-s + (−0.642 − 0.766i)17-s + ⋯
L(s)  = 1  + (0.984 − 0.173i)2-s + (0.597 − 0.802i)3-s + (0.939 − 0.342i)4-s + (−0.686 − 0.727i)5-s + (0.448 − 0.893i)6-s + (0.973 − 0.230i)7-s + (0.866 − 0.5i)8-s + (−0.286 − 0.957i)9-s + (−0.802 − 0.597i)10-s + (−0.802 + 0.597i)11-s + (0.286 − 0.957i)12-s + (0.230 + 0.973i)13-s + (0.918 − 0.396i)14-s + (−0.993 + 0.116i)15-s + (0.766 − 0.642i)16-s + (−0.642 − 0.766i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.139 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.139 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(109\)
Sign: $-0.139 - 0.990i$
Analytic conductor: \(11.7136\)
Root analytic conductor: \(11.7136\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{109} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 109,\ (1:\ ),\ -0.139 - 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.409550182 - 2.772043010i\)
\(L(\frac12)\) \(\approx\) \(2.409550182 - 2.772043010i\)
\(L(1)\) \(\approx\) \(1.922564733 - 1.144252175i\)
\(L(1)\) \(\approx\) \(1.922564733 - 1.144252175i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad109 \( 1 \)
good2 \( 1 + (0.984 - 0.173i)T \)
3 \( 1 + (0.597 - 0.802i)T \)
5 \( 1 + (-0.686 - 0.727i)T \)
7 \( 1 + (0.973 - 0.230i)T \)
11 \( 1 + (-0.802 + 0.597i)T \)
13 \( 1 + (0.230 + 0.973i)T \)
17 \( 1 + (-0.642 - 0.766i)T \)
19 \( 1 + (0.342 - 0.939i)T \)
23 \( 1 + (0.342 + 0.939i)T \)
29 \( 1 + (0.993 + 0.116i)T \)
31 \( 1 + (-0.973 - 0.230i)T \)
37 \( 1 + (0.727 + 0.686i)T \)
41 \( 1 + (0.866 + 0.5i)T \)
43 \( 1 + (-0.173 - 0.984i)T \)
47 \( 1 + (0.998 + 0.0581i)T \)
53 \( 1 + (-0.727 + 0.686i)T \)
59 \( 1 + (-0.116 - 0.993i)T \)
61 \( 1 + (0.0581 + 0.998i)T \)
67 \( 1 + (-0.230 + 0.973i)T \)
71 \( 1 + (-0.173 + 0.984i)T \)
73 \( 1 + (0.597 + 0.802i)T \)
79 \( 1 + (0.957 - 0.286i)T \)
83 \( 1 + (-0.396 - 0.918i)T \)
89 \( 1 + (0.893 + 0.448i)T \)
97 \( 1 + (-0.286 + 0.957i)T \)
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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

−30.045786836930414753013458924152, −28.54819684222971737540796523699, −27.21054959663415958587528023401, −26.540098786832110570035618393535, −25.39040913324387646451101221542, −24.37613822987010313252588025427, −23.259923658246441626623631556899, −22.282377162641711928692926426872, −21.35742607362110784807978693090, −20.53775194961099175247251048673, −19.51026157108042463677768004041, −18.0903043812637514875211360790, −16.39780213098875411538593469991, −15.46885091520924343807824311605, −14.78756174256623475675787501107, −13.95066838449723809576381536340, −12.543676039151406727149879808, −10.97423974979967912842683117247, −10.65628007882477445350961511699, −8.32908471481823144153445006218, −7.727137249453377008742464400103, −5.87930175903503679003847561547, −4.64894276485207181664561099744, −3.505817544717096161774173525435, −2.44549348318704677132611384454, 1.187748956114827985924361326696, 2.51084979462134512683269679338, 4.15443271952150782550002497424, 5.14410376605538516367758587570, 6.99995218393520328002750247523, 7.763444976000147319494612375621, 9.16027447548480540218621529404, 11.20188513881200868477400609261, 11.93319726961121695903655347167, 13.10082283268346451469894697282, 13.85592808514927221698469548469, 15.041357023256990761542256133709, 15.94926514630427057141263104929, 17.52140287178305447623043948917, 18.83067739008513655092863370756, 20.0739787152450576879900253672, 20.49482144087871581417793238702, 21.58591586927494204856659356259, 23.38821300188012668197210461377, 23.72195358997554278464418295331, 24.54951504618659608702147575365, 25.58010329972190694305232961332, 26.9041075419296762847400623502, 28.37180011189849979932501288429, 29.138984948097351325570694959024

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