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Properties

Label 1-109-109.27-r0-0-0
Degree $1$
Conductor $109$
Sign $0.878 + 0.477i$
Analytic cond. $0.506193$
Root an. cond. $0.506193$
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  + (−0.5 + 0.866i)2-s + (0.766 + 0.642i)3-s + (−0.5 − 0.866i)4-s + (0.173 − 0.984i)5-s + (−0.939 + 0.342i)6-s + (0.173 − 0.984i)7-s + 8-s + (0.173 + 0.984i)9-s + (0.766 + 0.642i)10-s + (0.766 − 0.642i)11-s + (0.173 − 0.984i)12-s + (0.173 − 0.984i)13-s + (0.766 + 0.642i)14-s + (0.766 − 0.642i)15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (0.766 + 0.642i)3-s + (−0.5 − 0.866i)4-s + (0.173 − 0.984i)5-s + (−0.939 + 0.342i)6-s + (0.173 − 0.984i)7-s + 8-s + (0.173 + 0.984i)9-s + (0.766 + 0.642i)10-s + (0.766 − 0.642i)11-s + (0.173 − 0.984i)12-s + (0.173 − 0.984i)13-s + (0.766 + 0.642i)14-s + (0.766 − 0.642i)15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.878 + 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.878 + 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(109\)
Sign: $0.878 + 0.477i$
Analytic conductor: \(0.506193\)
Root analytic conductor: \(0.506193\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{109} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 109,\ (0:\ ),\ 0.878 + 0.477i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.002984082 + 0.2547766626i\)
\(L(\frac12)\) \(\approx\) \(1.002984082 + 0.2547766626i\)
\(L(1)\) \(\approx\) \(1.001699792 + 0.2703053164i\)
\(L(1)\) \(\approx\) \(1.001699792 + 0.2703053164i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad109 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (0.766 + 0.642i)T \)
5 \( 1 + (0.173 - 0.984i)T \)
7 \( 1 + (0.173 - 0.984i)T \)
11 \( 1 + (0.766 - 0.642i)T \)
13 \( 1 + (0.173 - 0.984i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.766 + 0.642i)T \)
31 \( 1 + (0.173 + 0.984i)T \)
37 \( 1 + (0.173 + 0.984i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (-0.939 - 0.342i)T \)
53 \( 1 + (0.173 - 0.984i)T \)
59 \( 1 + (0.766 - 0.642i)T \)
61 \( 1 + (-0.939 + 0.342i)T \)
67 \( 1 + (0.173 + 0.984i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (0.766 - 0.642i)T \)
79 \( 1 + (0.173 + 0.984i)T \)
83 \( 1 + (0.766 + 0.642i)T \)
89 \( 1 + (-0.939 + 0.342i)T \)
97 \( 1 + (0.173 - 0.984i)T \)
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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

−29.612806399691649571905169155478, −28.5674431742855748707990426262, −27.438432947953172893207353607632, −26.29292606742423664369355774695, −25.64955394843528653948395720034, −24.6654635260050347916425111725, −23.07209352647409079080400797973, −21.92750401092238026225857464360, −21.143178470683402106757824493685, −19.792168703120371496828887973289, −19.09194864756573486761483742100, −18.199020732929952703774960763665, −17.50898330854436973798879823125, −15.53025602833696743408926574970, −14.34436966232228707547364057396, −13.428068096796695066803148490581, −12.03594494704071329224724054113, −11.354188694164068277537206554460, −9.60142014720229214970541923817, −8.996093732172688010628390033518, −7.569411727470530879689803093654, −6.49920553492569693349737501135, −4.135003732457777293525444711753, −2.65269189566166149716006938333, −1.929166101074548906033410803730, 1.358002502277285859067494774683, 3.86102500150440281511180031234, 4.88742304980313136320553926455, 6.364791247533764672834073750174, 8.14582744584535904897269458353, 8.50793187332065310380292645301, 9.882089501840016426049119273932, 10.70837746111141473682199467213, 12.94896575570510220773181159137, 13.96284661029283571564961655735, 14.83042073455959671594719153327, 16.1706524504562851377161627917, 16.71849866714684848840942341921, 17.789554352860248574417049728786, 19.51105197831603267852001488018, 20.00964852966272444663898007920, 21.16561918958165427213619182868, 22.52706878968271910311571914849, 23.81956920598819068235624163916, 24.733850929843457113180054345797, 25.48201700043999721831064668279, 26.62407168585563383719313942187, 27.35391653472460579872094437302, 28.104852665710087880794277307919, 29.4987427017404199247785793742

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