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Properties

Label 1-109-109.61-r0-0-0
Degree $1$
Conductor $109$
Sign $0.973 - 0.228i$
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  + (−0.173 + 0.984i)2-s + (0.396 − 0.918i)3-s + (−0.939 − 0.342i)4-s + (0.973 + 0.230i)5-s + (0.835 + 0.549i)6-s + (−0.286 − 0.957i)7-s + (0.5 − 0.866i)8-s + (−0.686 − 0.727i)9-s + (−0.396 + 0.918i)10-s + (−0.396 − 0.918i)11-s + (−0.686 + 0.727i)12-s + (0.286 + 0.957i)13-s + (0.993 − 0.116i)14-s + (0.597 − 0.802i)15-s + (0.766 + 0.642i)16-s + (−0.766 − 0.642i)17-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)2-s + (0.396 − 0.918i)3-s + (−0.939 − 0.342i)4-s + (0.973 + 0.230i)5-s + (0.835 + 0.549i)6-s + (−0.286 − 0.957i)7-s + (0.5 − 0.866i)8-s + (−0.686 − 0.727i)9-s + (−0.396 + 0.918i)10-s + (−0.396 − 0.918i)11-s + (−0.686 + 0.727i)12-s + (0.286 + 0.957i)13-s + (0.993 − 0.116i)14-s + (0.597 − 0.802i)15-s + (0.766 + 0.642i)16-s + (−0.766 − 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(109\)
Sign: $0.973 - 0.228i$
Analytic conductor: \(0.506193\)
Root analytic conductor: \(0.506193\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{109} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 109,\ (0:\ ),\ 0.973 - 0.228i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.053695304 - 0.1222057533i\)
\(L(\frac12)\) \(\approx\) \(1.053695304 - 0.1222057533i\)
\(L(1)\) \(\approx\) \(1.070019042 + 0.02446475861i\)
\(L(1)\) \(\approx\) \(1.070019042 + 0.02446475861i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad109 \( 1 \)
good2 \( 1 + (-0.173 + 0.984i)T \)
3 \( 1 + (0.396 - 0.918i)T \)
5 \( 1 + (0.973 + 0.230i)T \)
7 \( 1 + (-0.286 - 0.957i)T \)
11 \( 1 + (-0.396 - 0.918i)T \)
13 \( 1 + (0.286 + 0.957i)T \)
17 \( 1 + (-0.766 - 0.642i)T \)
19 \( 1 + (0.939 - 0.342i)T \)
23 \( 1 + (0.939 + 0.342i)T \)
29 \( 1 + (0.597 + 0.802i)T \)
31 \( 1 + (-0.286 + 0.957i)T \)
37 \( 1 + (-0.973 + 0.230i)T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (0.173 - 0.984i)T \)
47 \( 1 + (-0.893 - 0.448i)T \)
53 \( 1 + (-0.973 - 0.230i)T \)
59 \( 1 + (-0.597 + 0.802i)T \)
61 \( 1 + (0.893 - 0.448i)T \)
67 \( 1 + (0.286 - 0.957i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
73 \( 1 + (0.396 + 0.918i)T \)
79 \( 1 + (0.686 + 0.727i)T \)
83 \( 1 + (-0.993 + 0.116i)T \)
89 \( 1 + (-0.835 - 0.549i)T \)
97 \( 1 + (-0.686 + 0.727i)T \)
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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

−29.30812723124469366149864633088, −28.39538178281995888915799060012, −27.92267149456303532238633043890, −26.56571509134122144576737067513, −25.71991202787706373100365019824, −24.83957077185268244190543687558, −22.6897197914544382908431212205, −22.20430523467361497433784673707, −21.028906508483782453360630764405, −20.59236188233301978393639857268, −19.38387907134872595026816829473, −18.08173011560143023344581756241, −17.27559457464724065641106158051, −15.77327319753945056194355086644, −14.671555952248012085485727443052, −13.35501593968286688276295997765, −12.509502856884389504817621737155, −10.95236592163278275118045478559, −9.93551674222372351499995315189, −9.25282464692577552777057421425, −8.1890234529136138849826119531, −5.70162337523877502713959904575, −4.707436677281212611758376695417, −3.074168727202339345192582597277, −2.083134784043784997459138924105, 1.22799255562075019166177685815, 3.224613671342829836551783527547, 5.196434768515890001789773215471, 6.60790580572859311283210000945, 7.090586833129778198493103632979, 8.60977248607231144663120359933, 9.52575078295969620605166191647, 11.01223456747409385083147807769, 13.016490853405440982520657648139, 13.8096370390837938807508727530, 14.19046638342831377459744475748, 15.93214882443362509891628072450, 17.00661333129692874071005741819, 17.9487386413743308121865720415, 18.74898423330209320454368386129, 19.86224394702772266977546557655, 21.32198457749860791275141237207, 22.63390533171326638398071548682, 23.68078311276367346076099835245, 24.4346314683882747624409514622, 25.39864832780945078270539295946, 26.297960954503231260967484574, 26.84666644868757266704509120505, 28.878803883243813180541058139018, 29.25973506947895772300664054932

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