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Properties

Label 1-99-99.13-r1-0-0
Degree $1$
Conductor $99$
Sign $0.841 + 0.540i$
Analytic cond. $10.6390$
Root an. cond. $10.6390$
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  + (−0.913 + 0.406i)2-s + (0.669 − 0.743i)4-s + (0.913 + 0.406i)5-s + (0.978 + 0.207i)7-s + (−0.309 + 0.951i)8-s − 10-s + (0.104 − 0.994i)13-s + (−0.978 + 0.207i)14-s + (−0.104 − 0.994i)16-s + (0.809 − 0.587i)17-s + (−0.309 + 0.951i)19-s + (0.913 − 0.406i)20-s + (−0.5 − 0.866i)23-s + (0.669 + 0.743i)25-s + (0.309 + 0.951i)26-s + ⋯
L(s)  = 1  + (−0.913 + 0.406i)2-s + (0.669 − 0.743i)4-s + (0.913 + 0.406i)5-s + (0.978 + 0.207i)7-s + (−0.309 + 0.951i)8-s − 10-s + (0.104 − 0.994i)13-s + (−0.978 + 0.207i)14-s + (−0.104 − 0.994i)16-s + (0.809 − 0.587i)17-s + (−0.309 + 0.951i)19-s + (0.913 − 0.406i)20-s + (−0.5 − 0.866i)23-s + (0.669 + 0.743i)25-s + (0.309 + 0.951i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.841 + 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.841 + 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.841 + 0.540i$
Analytic conductor: \(10.6390\)
Root analytic conductor: \(10.6390\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 99,\ (1:\ ),\ 0.841 + 0.540i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.445123129 + 0.4245050615i\)
\(L(\frac12)\) \(\approx\) \(1.445123129 + 0.4245050615i\)
\(L(1)\) \(\approx\) \(0.9802076945 + 0.2014373693i\)
\(L(1)\) \(\approx\) \(0.9802076945 + 0.2014373693i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.913 + 0.406i)T \)
5 \( 1 + (0.913 + 0.406i)T \)
7 \( 1 + (0.978 + 0.207i)T \)
13 \( 1 + (0.104 - 0.994i)T \)
17 \( 1 + (0.809 - 0.587i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.978 + 0.207i)T \)
31 \( 1 + (-0.104 + 0.994i)T \)
37 \( 1 + (0.309 + 0.951i)T \)
41 \( 1 + (0.978 - 0.207i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (0.669 + 0.743i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.669 - 0.743i)T \)
61 \( 1 + (0.104 + 0.994i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (-0.309 - 0.951i)T \)
79 \( 1 + (-0.913 + 0.406i)T \)
83 \( 1 + (0.104 + 0.994i)T \)
89 \( 1 + T \)
97 \( 1 + (0.913 - 0.406i)T \)
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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

−29.60758863369507088703019647321, −28.39576290682724473746689307778, −27.8416333661430663106657112620, −26.52532212074913849869696163647, −25.71936767430954004329361045439, −24.641123981466374260685874938627, −23.69889331896927752492110860836, −21.61664926113318309146221309668, −21.29939796601401276475095366807, −20.17027824261583614218940688283, −19.003325701683874499962745716625, −17.77601729347473284499138003580, −17.21547967588588879260530735146, −16.10673232378416848889224280532, −14.50598682793591446183119094080, −13.272483998941945200847166167212, −11.93928758265984711878838879785, −10.87535374277884704360651439024, −9.69197585060954153766225686179, −8.71580492416542879787340274365, −7.512784909256122227248561684990, −6.011396980241493329635026458341, −4.301229500394305409185705999417, −2.291630459772462114243125477265, −1.171644198003991499643590319225, 1.24770342749134249994171805431, 2.64376000853978928569128000274, 5.18956984769428198625560646556, 6.19013160971701504164295106564, 7.62882370229942409934332445645, 8.651450087629370575462743189630, 10.04534423650994367030712217792, 10.75759069011931538411279395582, 12.223019046854847343230590433323, 14.09417899948602218464321716788, 14.73312379833827428409627218192, 16.088298536083362708762240271773, 17.336370319444515906635909118246, 18.03240736121062018247964219947, 18.85608719706282856142452591017, 20.42171110716094231265022300904, 21.134882971901200763689523980863, 22.59910193062214436835434489441, 23.88151980691085391324339465980, 25.09515390991774211175275988353, 25.44465167740494295606967303785, 26.88889304967153660270345843255, 27.54067456763087235515750720021, 28.696616174842789276716801332345, 29.68402743448315996927252883145

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