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Properties

Label 1-203-203.153-r0-0-0
Degree $1$
Conductor $203$
Sign $0.598 + 0.801i$
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  + (0.781 + 0.623i)2-s + (0.974 + 0.222i)3-s + (0.222 + 0.974i)4-s + (0.623 − 0.781i)5-s + (0.623 + 0.781i)6-s + (−0.433 + 0.900i)8-s + (0.900 + 0.433i)9-s + (0.974 − 0.222i)10-s + (−0.433 − 0.900i)11-s + i·12-s + (−0.900 + 0.433i)13-s + (0.781 − 0.623i)15-s + (−0.900 + 0.433i)16-s − i·17-s + (0.433 + 0.900i)18-s + (−0.974 + 0.222i)19-s + ⋯
L(s)  = 1  + (0.781 + 0.623i)2-s + (0.974 + 0.222i)3-s + (0.222 + 0.974i)4-s + (0.623 − 0.781i)5-s + (0.623 + 0.781i)6-s + (−0.433 + 0.900i)8-s + (0.900 + 0.433i)9-s + (0.974 − 0.222i)10-s + (−0.433 − 0.900i)11-s + i·12-s + (−0.900 + 0.433i)13-s + (0.781 − 0.623i)15-s + (−0.900 + 0.433i)16-s − i·17-s + (0.433 + 0.900i)18-s + (−0.974 + 0.222i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.598 + 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.598 + 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(203\)    =    \(7 \cdot 29\)
Sign: $0.598 + 0.801i$
Analytic conductor: \(0.942728\)
Root analytic conductor: \(0.942728\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{203} (153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 203,\ (0:\ ),\ 0.598 + 0.801i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.192102413 + 1.098563519i\)
\(L(\frac12)\) \(\approx\) \(2.192102413 + 1.098563519i\)
\(L(1)\) \(\approx\) \(1.942214161 + 0.7149133120i\)
\(L(1)\) \(\approx\) \(1.942214161 + 0.7149133120i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.781 + 0.623i)T \)
3 \( 1 + (0.974 + 0.222i)T \)
5 \( 1 + (0.623 - 0.781i)T \)
11 \( 1 + (-0.433 - 0.900i)T \)
13 \( 1 + (-0.900 + 0.433i)T \)
17 \( 1 - iT \)
19 \( 1 + (-0.974 + 0.222i)T \)
23 \( 1 + (0.623 + 0.781i)T \)
31 \( 1 + (-0.781 - 0.623i)T \)
37 \( 1 + (-0.433 + 0.900i)T \)
41 \( 1 + iT \)
43 \( 1 + (-0.781 + 0.623i)T \)
47 \( 1 + (-0.433 - 0.900i)T \)
53 \( 1 + (0.623 - 0.781i)T \)
59 \( 1 - T \)
61 \( 1 + (0.974 + 0.222i)T \)
67 \( 1 + (0.900 + 0.433i)T \)
71 \( 1 + (0.900 - 0.433i)T \)
73 \( 1 + (-0.781 + 0.623i)T \)
79 \( 1 + (0.433 - 0.900i)T \)
83 \( 1 + (0.222 + 0.974i)T \)
89 \( 1 + (-0.781 - 0.623i)T \)
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

−26.55404820373301486593199653026, −25.61273147688418774019359940628, −24.87026567045216070651350799323, −23.803656884709407108665964696559, −22.8131896350196876227035637023, −21.780171480014788719127025440147, −21.11117654557054247650477799028, −20.1146629781676522940727058546, −19.28861577876174990206584393959, −18.45092216511338899326767758843, −17.37220277562761043600594199054, −15.41259861467168600985009914440, −14.823237602387009561247587745579, −14.15375663809070052001669231382, −12.92504543558113771671979243888, −12.5041759548477889072786345552, −10.69117454204416103207493321169, −10.140651178068099093791996108207, −9.0063554638937785948634163931, −7.38450987282420658850734095178, −6.490561631174602723598381592024, −5.046591513555166593215750213743, −3.74789906072763690577710312635, −2.55598754316687451235111668344, −1.9111620476410660971459321781, 2.08470954342588712709692757334, 3.2167620248986167387964937127, 4.55858395025311170391231920903, 5.38095527918108273265283056729, 6.808183914253053487436447600291, 7.99978406134749135890053369266, 8.84902184999860747008509482698, 9.85066196486975365830674890532, 11.51104180464400454465071724674, 12.86342674484871825308355286170, 13.450467838628902224019722140023, 14.30961315372675167920525815389, 15.24325586738281301910329458889, 16.34122683998405449149961947167, 16.93421073299828577524971653958, 18.3468359247300166228355135290, 19.63446017211196120043084253611, 20.66924742072818504322252929568, 21.37688893316852193495835089805, 21.96474928076606235566647186821, 23.47880436641721654526355435184, 24.390156235175537430034737584679, 24.96567540410400142847397211492, 25.79980370075114386726818968833, 26.69939298772552323261901622500

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