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Properties

Label 1-103-103.42-r1-0-0
Degree $1$
Conductor $103$
Sign $-0.947 - 0.319i$
Analytic cond. $11.0688$
Root an. cond. $11.0688$
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  + (−0.982 − 0.183i)2-s + (−0.0922 + 0.995i)3-s + (0.932 + 0.361i)4-s + (0.602 − 0.798i)5-s + (0.273 − 0.961i)6-s + (−0.850 + 0.526i)7-s + (−0.850 − 0.526i)8-s + (−0.982 − 0.183i)9-s + (−0.739 + 0.673i)10-s + (0.982 + 0.183i)11-s + (−0.445 + 0.895i)12-s + (−0.850 + 0.526i)13-s + (0.932 − 0.361i)14-s + (0.739 + 0.673i)15-s + (0.739 + 0.673i)16-s + (−0.273 − 0.961i)17-s + ⋯
L(s)  = 1  + (−0.982 − 0.183i)2-s + (−0.0922 + 0.995i)3-s + (0.932 + 0.361i)4-s + (0.602 − 0.798i)5-s + (0.273 − 0.961i)6-s + (−0.850 + 0.526i)7-s + (−0.850 − 0.526i)8-s + (−0.982 − 0.183i)9-s + (−0.739 + 0.673i)10-s + (0.982 + 0.183i)11-s + (−0.445 + 0.895i)12-s + (−0.850 + 0.526i)13-s + (0.932 − 0.361i)14-s + (0.739 + 0.673i)15-s + (0.739 + 0.673i)16-s + (−0.273 − 0.961i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(103\)
Sign: $-0.947 - 0.319i$
Analytic conductor: \(11.0688\)
Root analytic conductor: \(11.0688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{103} (42, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 103,\ (1:\ ),\ -0.947 - 0.319i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01783961245 + 0.1085791553i\)
\(L(\frac12)\) \(\approx\) \(0.01783961245 + 0.1085791553i\)
\(L(1)\) \(\approx\) \(0.5144025965 + 0.1146421639i\)
\(L(1)\) \(\approx\) \(0.5144025965 + 0.1146421639i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad103 \( 1 \)
good2 \( 1 + (-0.982 - 0.183i)T \)
3 \( 1 + (-0.0922 + 0.995i)T \)
5 \( 1 + (0.602 - 0.798i)T \)
7 \( 1 + (-0.850 + 0.526i)T \)
11 \( 1 + (0.982 + 0.183i)T \)
13 \( 1 + (-0.850 + 0.526i)T \)
17 \( 1 + (-0.273 - 0.961i)T \)
19 \( 1 + (0.0922 + 0.995i)T \)
23 \( 1 + (-0.982 + 0.183i)T \)
29 \( 1 + (-0.602 + 0.798i)T \)
31 \( 1 + (-0.739 - 0.673i)T \)
37 \( 1 + (-0.445 + 0.895i)T \)
41 \( 1 + (-0.602 - 0.798i)T \)
43 \( 1 + (-0.445 - 0.895i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.0922 - 0.995i)T \)
59 \( 1 + (-0.850 - 0.526i)T \)
61 \( 1 + (-0.273 - 0.961i)T \)
67 \( 1 + (0.850 + 0.526i)T \)
71 \( 1 + (0.602 + 0.798i)T \)
73 \( 1 + (0.602 + 0.798i)T \)
79 \( 1 + (-0.602 + 0.798i)T \)
83 \( 1 + (-0.850 + 0.526i)T \)
89 \( 1 + (-0.932 + 0.361i)T \)
97 \( 1 + (-0.273 + 0.961i)T \)
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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

−29.07574803850608718234089201278, −28.10701933790723492220050995817, −26.60766965975673257646516628061, −25.95977522610591158440813084144, −25.00886341086808166890932557177, −24.15614366384306361914148149476, −22.843231019737476934820971250127, −21.81424776428107809471539605755, −19.73924320755251474116857068574, −19.63843092607289631275812395959, −18.31393189591131353132272850122, −17.44861600447158582208615722199, −16.7295902742104566721550323385, −15.05587807376766002447649647451, −13.99090442366061007288265135097, −12.6965521915913139503202112422, −11.35395054225252688463785087422, −10.248880085445063755110300213250, −9.14074364816620749085903351672, −7.622916913176868577538871706210, −6.694342199955469772397421791936, −5.988069279924308246771961183408, −3.06986652767887674701106982658, −1.74913736732033091216752771334, −0.06169825398935666173849898083, 2.0346881694698581852731368682, 3.67282535301774645533903898760, 5.39488754972682742076652258312, 6.667566557814188375711334975744, 8.55659897159850080475529980071, 9.56528623907787925293517483573, 9.83989806035153209892984269412, 11.601045135001387242777535879235, 12.42581926923324327778794228037, 14.26909553617596820855426721892, 15.65189602932557534957872447961, 16.57823729569423453526943410180, 17.10745215297782189670042186689, 18.50600871395143020818751965552, 19.87052010821713235279480595026, 20.479403688881910846893000625686, 21.723222583901961112391371391802, 22.35394229119225603552917737774, 24.33563389786112378822822690975, 25.30837628057776928758548474352, 26.020062639016866019202271090, 27.31169933502466970046220626624, 27.8882395044627494836384830711, 29.03272325486951216071308614432, 29.40852531638170070471178292641

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