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Properties

Label 1-203-203.18-r1-0-0
Degree $1$
Conductor $203$
Sign $-0.470 - 0.882i$
Analytic cond. $21.8153$
Root an. cond. $21.8153$
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  + (−0.149 − 0.988i)2-s + (−0.680 − 0.733i)3-s + (−0.955 + 0.294i)4-s + (0.988 − 0.149i)5-s + (−0.623 + 0.781i)6-s + (0.433 + 0.900i)8-s + (−0.0747 + 0.997i)9-s + (−0.294 − 0.955i)10-s + (−0.997 + 0.0747i)11-s + (0.866 + 0.5i)12-s + (0.900 + 0.433i)13-s + (−0.781 − 0.623i)15-s + (0.826 − 0.563i)16-s + (0.866 − 0.5i)17-s + (0.997 − 0.0747i)18-s + (0.680 − 0.733i)19-s + ⋯
L(s)  = 1  + (−0.149 − 0.988i)2-s + (−0.680 − 0.733i)3-s + (−0.955 + 0.294i)4-s + (0.988 − 0.149i)5-s + (−0.623 + 0.781i)6-s + (0.433 + 0.900i)8-s + (−0.0747 + 0.997i)9-s + (−0.294 − 0.955i)10-s + (−0.997 + 0.0747i)11-s + (0.866 + 0.5i)12-s + (0.900 + 0.433i)13-s + (−0.781 − 0.623i)15-s + (0.826 − 0.563i)16-s + (0.866 − 0.5i)17-s + (0.997 − 0.0747i)18-s + (0.680 − 0.733i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.470 - 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.470 - 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(203\)    =    \(7 \cdot 29\)
Sign: $-0.470 - 0.882i$
Analytic conductor: \(21.8153\)
Root analytic conductor: \(21.8153\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{203} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 203,\ (1:\ ),\ -0.470 - 0.882i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7515189207 - 1.251622879i\)
\(L(\frac12)\) \(\approx\) \(0.7515189207 - 1.251622879i\)
\(L(1)\) \(\approx\) \(0.7288657583 - 0.5953622114i\)
\(L(1)\) \(\approx\) \(0.7288657583 - 0.5953622114i\)

Euler product

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

−26.64198499235139523648994233768, −26.07320410678591267770178021834, −25.17600699804528251458000766988, −24.09060659240209276352929555239, −23.03131297388942782737532290502, −22.50977342107933309849577432060, −21.30118130176213431611521281251, −20.726472106373332583794077831185, −18.70607905418273696514729181299, −18.09933971615191796921792166448, −17.17053990349371868681475516087, −16.375200437320302127221729764150, −15.51837709758502349619525076235, −14.52310065799811088504770506312, −13.50678701248411080376838506517, −12.43908912823870625625001759398, −10.620740257720792442245162273068, −10.15798459772576972194825969891, −9.0435128096429617405288819710, −7.860445768695949876215394569974, −6.3195979912314750333621341530, −5.72444131467539462386108761032, −4.806536528058605913337487911625, −3.3018342003551748692600708141, −1.000916073292894727163473653042, 0.78559938755933213813172046256, 1.83663881571400159425912774720, 3.03883475663972519519100890707, 4.95670436760102080732762706459, 5.66158081103992775779057417817, 7.164473854534847410210066919348, 8.43845312737661333303346172724, 9.65229091291794154800430133758, 10.61005188258395913537076103600, 11.53117692140526785077515064260, 12.56028299288721044681210534303, 13.495403466006668354903437966972, 13.927690484085059687297511685208, 15.98163578452448455281164972607, 17.117938499993390158837658271248, 17.9417817833448850173752920148, 18.50141302505275682075033215583, 19.48733470252174925379424164758, 20.86466444304649558259504338179, 21.333825525873965121972306671602, 22.5098272011821617918417918297, 23.2682128350303874432084215734, 24.24567791058072132014093675292, 25.51623724214392433287434701623, 26.23816226018225835336512812194

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