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Properties

Label 1-109-109.64-r0-0-0
Degree $1$
Conductor $109$
Sign $-0.997 + 0.0639i$
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  − 2-s + (−0.5 − 0.866i)3-s + 4-s + (−0.5 − 0.866i)5-s + (0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s − 8-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)10-s + (0.5 − 0.866i)11-s + (−0.5 − 0.866i)12-s + (0.5 + 0.866i)13-s + (0.5 + 0.866i)14-s + (−0.5 + 0.866i)15-s + 16-s − 17-s + ⋯
L(s)  = 1  − 2-s + (−0.5 − 0.866i)3-s + 4-s + (−0.5 − 0.866i)5-s + (0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s − 8-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)10-s + (0.5 − 0.866i)11-s + (−0.5 − 0.866i)12-s + (0.5 + 0.866i)13-s + (0.5 + 0.866i)14-s + (−0.5 + 0.866i)15-s + 16-s − 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0639i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(109\)
Sign: $-0.997 + 0.0639i$
Analytic conductor: \(0.506193\)
Root analytic conductor: \(0.506193\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{109} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 109,\ (0:\ ),\ -0.997 + 0.0639i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.009139100024 - 0.2857087708i\)
\(L(\frac12)\) \(\approx\) \(0.009139100024 - 0.2857087708i\)
\(L(1)\) \(\approx\) \(0.3539828748 - 0.2578565877i\)
\(L(1)\) \(\approx\) \(0.3539828748 - 0.2578565877i\)

Euler product

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

−29.80570878103636874292542465349, −28.68876595376886999895000510072, −27.75984353746352422624538316155, −27.32015809201151722089906801535, −25.91146010397047704252809040382, −25.59753196502485049302271628651, −23.91179416516920428560936412963, −22.56085570755880606855384211452, −21.969203975922329408372079870057, −20.52177959876379996236321587784, −19.62908693869336478126782021339, −18.36543879544339968752929065175, −17.65773686978647765736241071261, −16.35460723082490699456113111385, −15.303647482305156173059495089765, −14.98438876994143293892250632343, −12.46441397738332229021290923420, −11.4253585921731586385016519626, −10.52692226084101669178412091360, −9.543089251048539351180780694500, −8.41406599794053794906199947636, −6.813026455512236477357489851244, −5.895350136190443929980838982259, −3.88527639349470853708360480026, −2.489714583941072471033296079, 0.392744232035325418770258526181, 1.810180296080187918053780704768, 3.977629153100466067164323879988, 6.04296793004238024758516092377, 6.96256030418118046954028520402, 8.17957382230362473884793439945, 9.101207979143254624951280391575, 10.78693989045124156512575272901, 11.59736762770264704960933690568, 12.735428134854227634683818046827, 13.88111551139937939177119821906, 15.873351116522518228621444444192, 16.66794440098146740530087023067, 17.29989591035947135304776993393, 18.67657341905620044646618038771, 19.5239859227038305321522997474, 20.172108399936654662580756293233, 21.65243375030478554550125066610, 23.31673776038384727064096605189, 24.02053397203927394167330998561, 24.81849026818071786400564386779, 26.041434939440388047806520057776, 27.07185612254392612485278473156, 28.16060031117600827324872089557, 28.88156302898669702817944641307

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