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Properties

Label 1-203-203.149-r0-0-0
Degree $1$
Conductor $203$
Sign $0.932 - 0.360i$
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  + (−0.0747 − 0.997i)2-s + (−0.365 − 0.930i)3-s + (−0.988 + 0.149i)4-s + (0.0747 + 0.997i)5-s + (−0.900 + 0.433i)6-s + (0.222 + 0.974i)8-s + (−0.733 + 0.680i)9-s + (0.988 − 0.149i)10-s + (0.733 + 0.680i)11-s + (0.5 + 0.866i)12-s + (−0.222 + 0.974i)13-s + (0.900 − 0.433i)15-s + (0.955 − 0.294i)16-s + (0.5 − 0.866i)17-s + (0.733 + 0.680i)18-s + (−0.365 + 0.930i)19-s + ⋯
L(s)  = 1  + (−0.0747 − 0.997i)2-s + (−0.365 − 0.930i)3-s + (−0.988 + 0.149i)4-s + (0.0747 + 0.997i)5-s + (−0.900 + 0.433i)6-s + (0.222 + 0.974i)8-s + (−0.733 + 0.680i)9-s + (0.988 − 0.149i)10-s + (0.733 + 0.680i)11-s + (0.5 + 0.866i)12-s + (−0.222 + 0.974i)13-s + (0.900 − 0.433i)15-s + (0.955 − 0.294i)16-s + (0.5 − 0.866i)17-s + (0.733 + 0.680i)18-s + (−0.365 + 0.930i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(203\)    =    \(7 \cdot 29\)
Sign: $0.932 - 0.360i$
Analytic conductor: \(0.942728\)
Root analytic conductor: \(0.942728\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{203} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 203,\ (0:\ ),\ 0.932 - 0.360i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8056166550 - 0.1501619753i\)
\(L(\frac12)\) \(\approx\) \(0.8056166550 - 0.1501619753i\)
\(L(1)\) \(\approx\) \(0.7688722726 - 0.2936536482i\)
\(L(1)\) \(\approx\) \(0.7688722726 - 0.2936536482i\)

Euler product

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

−27.001375568413731331648704021551, −25.91480407030751743302687392565, −25.028178685967808459275951811438, −24.1052023219322201060501018934, −23.27511664090058996059409320105, −22.180294259030197638478681087661, −21.52831309904370853351281657731, −20.32338586284790228088121525809, −19.28411686999511234785226133217, −17.7814392825467351809742924392, −16.92081552812116736862694130073, −16.538177962803481235252559404626, −15.365211015746679422931401210286, −14.71207176333131234467690809548, −13.366694545744303059704412792121, −12.40627900909162366340557296078, −10.96213734091776468453696464389, −9.75613941083614011959643048416, −8.902768193169977539122625746516, −8.10374756344196166589616042231, −6.41161321340593468291473308341, −5.48413436477478241225885459378, −4.63163612254242018315258530590, −3.52498987326529242408251704857, −0.74882046608124194092722804781, 1.49875673945233702067835783631, 2.48997747001878274083775959731, 3.82125015056696278502990153623, 5.34408793613864465066417878826, 6.73407472286707153269305249603, 7.58977784777841929515118808486, 9.08279710513445265012143924495, 10.149833545438860225073932629558, 11.32318961472996858243277040695, 11.884456827693941108400641406248, 12.91335411094173094241217317287, 14.10451935789067980548259909577, 14.59173397168989425944680335570, 16.63953912035142853406115920668, 17.568978200267052317434270032580, 18.43234476634950989713857025031, 19.07018877714980763596016395168, 19.86512282358294557388268513916, 21.15459575118772918837976509196, 22.16348292072688554575835153951, 22.940283707827458339649823278630, 23.53019922852373079539731522982, 25.04283494924405315101991183682, 25.83706417636784749870690795588, 27.06759972490340644416105143483

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