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Properties

Label 1-203-203.47-r0-0-0
Degree $1$
Conductor $203$
Sign $0.851 - 0.524i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.930 + 0.365i)2-s + (0.294 − 0.955i)3-s + (0.733 + 0.680i)4-s + (0.365 − 0.930i)5-s + (0.623 − 0.781i)6-s + (0.433 + 0.900i)8-s + (−0.826 − 0.563i)9-s + (0.680 − 0.733i)10-s + (0.563 + 0.826i)11-s + (0.866 − 0.5i)12-s + (−0.900 − 0.433i)13-s + (−0.781 − 0.623i)15-s + (0.0747 + 0.997i)16-s + (0.866 + 0.5i)17-s + (−0.563 − 0.826i)18-s + (−0.294 − 0.955i)19-s + ⋯
L(s)  = 1  + (0.930 + 0.365i)2-s + (0.294 − 0.955i)3-s + (0.733 + 0.680i)4-s + (0.365 − 0.930i)5-s + (0.623 − 0.781i)6-s + (0.433 + 0.900i)8-s + (−0.826 − 0.563i)9-s + (0.680 − 0.733i)10-s + (0.563 + 0.826i)11-s + (0.866 − 0.5i)12-s + (−0.900 − 0.433i)13-s + (−0.781 − 0.623i)15-s + (0.0747 + 0.997i)16-s + (0.866 + 0.5i)17-s + (−0.563 − 0.826i)18-s + (−0.294 − 0.955i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(203\)    =    \(7 \cdot 29\)
Sign: $0.851 - 0.524i$
Analytic conductor: \(0.942728\)
Root analytic conductor: \(0.942728\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{203} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 203,\ (0:\ ),\ 0.851 - 0.524i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.163700747 - 0.6136306451i\)
\(L(\frac12)\) \(\approx\) \(2.163700747 - 0.6136306451i\)
\(L(1)\) \(\approx\) \(1.896671327 - 0.3031659180i\)
\(L(1)\) \(\approx\) \(1.896671327 - 0.3031659180i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.930 + 0.365i)T \)
3 \( 1 + (0.294 - 0.955i)T \)
5 \( 1 + (0.365 - 0.930i)T \)
11 \( 1 + (0.563 + 0.826i)T \)
13 \( 1 + (-0.900 - 0.433i)T \)
17 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (-0.294 - 0.955i)T \)
23 \( 1 + (-0.988 - 0.149i)T \)
31 \( 1 + (0.149 + 0.988i)T \)
37 \( 1 + (0.563 - 0.826i)T \)
41 \( 1 + iT \)
43 \( 1 + (0.781 + 0.623i)T \)
47 \( 1 + (-0.997 + 0.0747i)T \)
53 \( 1 + (-0.988 + 0.149i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (0.680 + 0.733i)T \)
67 \( 1 + (-0.0747 + 0.997i)T \)
71 \( 1 + (0.900 + 0.433i)T \)
73 \( 1 + (-0.930 + 0.365i)T \)
79 \( 1 + (-0.563 + 0.826i)T \)
83 \( 1 + (0.222 - 0.974i)T \)
89 \( 1 + (-0.930 - 0.365i)T \)
97 \( 1 + (-0.974 - 0.222i)T \)
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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

−27.02230151551750460552830397355, −25.89779381847667515622241437752, −25.09683607295377391539276832205, −23.970536584977327909721986191044, −22.67900984788393840065693586863, −22.17816858143604215144605880005, −21.41645325548375159086507666243, −20.61868210478699729895125067345, −19.43632611805573790976046414113, −18.76017075898041016830506751558, −17.007272385382268878782834039499, −16.134223906608765358603717713689, −14.954560827819126329498052053758, −14.3065020146491246403415589433, −13.73880947781844238526326011417, −12.05161348917625058087179307465, −11.22572954798323717642482182608, −10.17352335504992619235652071756, −9.544874976270171671798612527579, −7.76730865693887323447915885046, −6.32756980932945166285959333313, −5.44510961427650948974976526011, −4.07911965451080797567760775008, −3.22186699322528848289651651970, −2.12994908152374829627491687446, 1.55990731855566502390005349441, 2.70739775065514377289058279982, 4.27996693978169811980614457498, 5.40782829444326984758923799259, 6.46684005477879812143382490977, 7.53666707006130184413558330027, 8.44543022655644428737421936303, 9.773667483994306881198525701404, 11.59295533832412010571586710341, 12.58364749068482363434352676626, 12.85534544126819946953936934480, 14.15964516922579021361157659651, 14.778028960457591131263423766625, 16.12700182477749708590765274652, 17.27362451511263474300730643471, 17.730158015318243741816957915898, 19.60812170720485800236201575488, 20.03801534442343188210756470694, 21.123214747486644272612145627609, 22.139016503060293388608361395352, 23.26645176731453089782282305751, 23.98678892764608806905609407076, 24.846825970572361506484609715497, 25.33620903937052276483353881299, 26.28726513576202660437020526902

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