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Properties

Label 1-109-109.59-r1-0-0
Degree $1$
Conductor $109$
Sign $0.968 + 0.249i$
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  + (0.342 − 0.939i)2-s + (−0.686 − 0.727i)3-s + (−0.766 − 0.642i)4-s + (0.893 + 0.448i)5-s + (−0.918 + 0.396i)6-s + (−0.835 + 0.549i)7-s + (−0.866 + 0.5i)8-s + (−0.0581 + 0.998i)9-s + (0.727 − 0.686i)10-s + (0.727 + 0.686i)11-s + (0.0581 + 0.998i)12-s + (0.549 + 0.835i)13-s + (0.230 + 0.973i)14-s + (−0.286 − 0.957i)15-s + (0.173 + 0.984i)16-s + (−0.984 + 0.173i)17-s + ⋯
L(s)  = 1  + (0.342 − 0.939i)2-s + (−0.686 − 0.727i)3-s + (−0.766 − 0.642i)4-s + (0.893 + 0.448i)5-s + (−0.918 + 0.396i)6-s + (−0.835 + 0.549i)7-s + (−0.866 + 0.5i)8-s + (−0.0581 + 0.998i)9-s + (0.727 − 0.686i)10-s + (0.727 + 0.686i)11-s + (0.0581 + 0.998i)12-s + (0.549 + 0.835i)13-s + (0.230 + 0.973i)14-s + (−0.286 − 0.957i)15-s + (0.173 + 0.984i)16-s + (−0.984 + 0.173i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.968 + 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.968 + 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(109\)
Sign: $0.968 + 0.249i$
Analytic conductor: \(11.7136\)
Root analytic conductor: \(11.7136\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{109} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 109,\ (1:\ ),\ 0.968 + 0.249i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.089191698 + 0.1377975688i\)
\(L(\frac12)\) \(\approx\) \(1.089191698 + 0.1377975688i\)
\(L(1)\) \(\approx\) \(0.8712402772 - 0.3132943260i\)
\(L(1)\) \(\approx\) \(0.8712402772 - 0.3132943260i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad109 \( 1 \)
good2 \( 1 + (0.342 - 0.939i)T \)
3 \( 1 + (-0.686 - 0.727i)T \)
5 \( 1 + (0.893 + 0.448i)T \)
7 \( 1 + (-0.835 + 0.549i)T \)
11 \( 1 + (0.727 + 0.686i)T \)
13 \( 1 + (0.549 + 0.835i)T \)
17 \( 1 + (-0.984 + 0.173i)T \)
19 \( 1 + (-0.642 - 0.766i)T \)
23 \( 1 + (-0.642 + 0.766i)T \)
29 \( 1 + (0.286 - 0.957i)T \)
31 \( 1 + (0.835 + 0.549i)T \)
37 \( 1 + (0.448 + 0.893i)T \)
41 \( 1 + (-0.866 - 0.5i)T \)
43 \( 1 + (0.939 + 0.342i)T \)
47 \( 1 + (-0.802 + 0.597i)T \)
53 \( 1 + (-0.448 + 0.893i)T \)
59 \( 1 + (-0.957 + 0.286i)T \)
61 \( 1 + (-0.597 + 0.802i)T \)
67 \( 1 + (-0.549 + 0.835i)T \)
71 \( 1 + (0.939 - 0.342i)T \)
73 \( 1 + (-0.686 + 0.727i)T \)
79 \( 1 + (0.998 + 0.0581i)T \)
83 \( 1 + (-0.973 + 0.230i)T \)
89 \( 1 + (0.396 + 0.918i)T \)
97 \( 1 + (-0.0581 - 0.998i)T \)
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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

−29.25400572299058068340168076105, −28.07339146280492199204983879791, −27.038597058521825642409643354285, −26.11595946558630340994500585358, −25.15221109885008154485203797859, −24.11622708372910128309611159444, −22.90508916848978442534043883443, −22.26291874440458601356562048862, −21.34346142973118366570341763746, −20.1795311699911757540787753817, −18.28743209671741165842298621217, −17.268052650069675586130055094872, −16.55350538303501685185002099353, −15.81297125640084638578925965133, −14.42822182107363494028312344630, −13.34138070051856271216288658852, −12.38491433214222282374170053978, −10.638976523316161455703315083162, −9.53687129777752509484934739029, −8.52046526572064967475303804905, −6.45526590319375265846958832770, −6.03317131741804389498805799852, −4.63452899531221766523745111761, −3.4805500310543337681762575408, −0.46688056891908990611484697495, 1.572086451065283174812933272786, 2.59750145367478617275864392473, 4.481500903699168997638545616260, 6.06096741033736007958898458462, 6.61672093371093767179083711331, 8.94158345860059848299090951829, 9.96072131857722505838066027866, 11.18567805045403335648107729624, 12.129734338304274173890378397112, 13.21503174109781893722985327831, 13.88943026474686462898438114177, 15.418284081464362872261392971011, 17.19403545107929943382002243421, 17.96359411048993280504187412740, 18.977296439945707561394318437245, 19.729783520102714017393405326214, 21.372382450031647278808773106100, 22.14277415765989683191034535888, 22.81374018745544481106361986740, 23.953639297073760815110400556158, 25.160389873840217604098174929038, 26.190754943520293174622574014323, 27.869812740231009385415470640176, 28.64888265807450223037850398059, 29.216217640051903346069066445278

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