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Properties

Label 1-203-203.26-r0-0-0
Degree $1$
Conductor $203$
Sign $-0.933 + 0.359i$
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  + (−0.563 + 0.826i)2-s + (0.149 + 0.988i)3-s + (−0.365 − 0.930i)4-s + (0.826 + 0.563i)5-s + (−0.900 − 0.433i)6-s + (0.974 + 0.222i)8-s + (−0.955 + 0.294i)9-s + (−0.930 + 0.365i)10-s + (−0.294 + 0.955i)11-s + (0.866 − 0.5i)12-s + (−0.222 − 0.974i)13-s + (−0.433 + 0.900i)15-s + (−0.733 + 0.680i)16-s + (0.866 + 0.5i)17-s + (0.294 − 0.955i)18-s + (−0.149 + 0.988i)19-s + ⋯
L(s)  = 1  + (−0.563 + 0.826i)2-s + (0.149 + 0.988i)3-s + (−0.365 − 0.930i)4-s + (0.826 + 0.563i)5-s + (−0.900 − 0.433i)6-s + (0.974 + 0.222i)8-s + (−0.955 + 0.294i)9-s + (−0.930 + 0.365i)10-s + (−0.294 + 0.955i)11-s + (0.866 − 0.5i)12-s + (−0.222 − 0.974i)13-s + (−0.433 + 0.900i)15-s + (−0.733 + 0.680i)16-s + (0.866 + 0.5i)17-s + (0.294 − 0.955i)18-s + (−0.149 + 0.988i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.933 + 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.933 + 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(203\)    =    \(7 \cdot 29\)
Sign: $-0.933 + 0.359i$
Analytic conductor: \(0.942728\)
Root analytic conductor: \(0.942728\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{203} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 203,\ (0:\ ),\ -0.933 + 0.359i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1667651706 + 0.8959772339i\)
\(L(\frac12)\) \(\approx\) \(0.1667651706 + 0.8959772339i\)
\(L(1)\) \(\approx\) \(0.5786346973 + 0.6580657984i\)
\(L(1)\) \(\approx\) \(0.5786346973 + 0.6580657984i\)

Euler product

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

−26.25186019235323044443860232467, −25.69135489248635457973772847945, −24.56843846903222970869907757169, −23.83700563312387334648238031921, −22.439477643449839039073750144295, −21.40813080423455351155126694738, −20.68012354008732283954869789466, −19.63330723720132726874409659144, −18.768194785588197142978063738601, −18.08508917391182710306081721145, −17.00786479101296069378318194697, −16.39704533102107992732575047961, −14.22754815880739475625991891996, −13.54952662582179815095236622906, −12.668921031746813652932807047617, −11.78055774538114165607260715394, −10.698864731633479240915574853111, −9.29974928631369226124391708325, −8.7013244138812364935056725915, −7.528341825725387247857012149646, −6.284475995959532506003267154486, −4.862193227288416001937278464025, −3.06797489466321150687986347918, −2.04723350912170592885335942455, −0.84600357175642181246356766870, 1.9677404060457106123493838917, 3.60307722562742342796420979022, 5.221481887527043389092308435593, 5.788951901929876158683239339511, 7.26834041186373021632195837912, 8.30282476475157889461420809527, 9.71860099234772246358474495050, 10.00811707565121561735863119716, 10.983779018741166524333944226645, 12.862498328620529950105758389164, 14.225697921400016888432438773769, 14.81370855852579366401885926542, 15.61335999079981113060158722342, 16.78620581320661643430461887946, 17.5064366676598592123534125590, 18.37602317665444918829625408340, 19.613307727673595899732004490022, 20.62534006529940909699130267455, 21.6546144367353294703380578177, 22.71335197754592958618996666617, 23.29525475084215622777474484977, 24.95904453142841051518121972203, 25.551922090162645003575720988029, 26.12908442908323967710248857924, 27.217592792541810630162983049943

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