↑

Properties

Label 1-203-203.104-r0-0-0
Degree $1$
Conductor $203$
Sign $0.981 - 0.189i$
Analytic cond. $0.942728$
Root an. cond. $0.942728$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s + i·3-s − 4-s + 5-s + 6-s + i·8-s − 9-s − i·10-s − i·11-s − i·12-s + 13-s + i·15-s + 16-s + i·17-s + i·18-s + i·19-s + ⋯
L(s)  = 1  − i·2-s + i·3-s − 4-s + 5-s + 6-s + i·8-s − 9-s − i·10-s − i·11-s − i·12-s + 13-s + i·15-s + 16-s + i·17-s + i·18-s + i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(203\)    =    \(7 \cdot 29\)
Sign: $0.981 - 0.189i$
Analytic conductor: \(0.942728\)
Root analytic conductor: \(0.942728\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{203} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 203,\ (0:\ ),\ 0.981 - 0.189i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.235092697 - 0.1178458416i\)
\(L(\frac12)\) \(\approx\) \(1.235092697 - 0.1178458416i\)
\(L(1)\) \(\approx\) \(1.114759108 - 0.1589645999i\)
\(L(1)\) \(\approx\) \(1.114759108 - 0.1589645999i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
29 \( 1 \)
good2 \( 1 \)
3 \( 1 + T \)
5 \( 1 - iT \)
11 \( 1 - T \)
13 \( 1 + T \)
17 \( 1 + T \)
19 \( 1 \)
23 \( 1 + iT \)
31 \( 1 - iT \)
37 \( 1 - iT \)
41 \( 1 - iT \)
43 \( 1 + T \)
47 \( 1 \)
53 \( 1 + iT \)
59 \( 1 + T \)
61 \( 1 + iT \)
67 \( 1 + iT \)
71 \( 1 + iT \)
73 \( 1 - T \)
79 \( 1 \)
83 \( 1 - T \)
89 \( 1 + T \)
97 \( 1 - T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−26.37621823111337287651851695130, −25.65904260434586078077578946105, −25.03429227814489170217217328760, −24.29694866075388112468217974462, −23.1344152197211054282458245607, −22.6502873160483462806481103135, −21.31174296558984568636631740049, −20.15142720155010561868416658054, −18.78805112627871965292542277868, −18.01422467861635286040057041027, −17.48619915320881324353791353888, −16.462863551401928896809507824858, −15.1639380967305447603401934668, −14.159338671378990286478155540258, −13.32514153680344806946887931826, −12.74582125997798947435054261279, −11.18140006791734424306667147364, −9.59746396716319087714923968873, −8.84410472368437623965615895839, −7.518363158222953189574485072752, −6.71531948985173915744662031289, −5.81277953140279443060113461526, −4.717498845290160424467641251383, −2.75186876910742530628235247268, −1.196239632333860290373525678051, 1.45820906732311289353615574635, 2.99946050453936037411352023546, 3.8783361761317691912399305109, 5.28386359162067972703056569536, 6.062217489520099951326717148872, 8.51851114206526167452792083242, 8.99564952124481047500075409763, 10.404097298853714035641633647198, 10.61865899604381811857230497997, 11.91545818594227205320324768952, 13.28285326700583934703830611553, 13.98072642110741550851861250266, 15.00285633616365368322809140993, 16.4897825412364569126125978855, 17.211795370044275877553928531252, 18.34239625834727819470575395922, 19.29289768481011911240645756294, 20.560106550587517316938652136987, 21.156833241023241548489676441762, 21.7610396228224808736021737384, 22.65168414586636619818333228740, 23.67512035808519251503854629446, 25.24132758741739438731367874791, 26.13116896565023354813358549174, 26.968373077451022167049210414044

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