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Properties

Label 1-103-103.18-r0-0-0
Degree $1$
Conductor $103$
Sign $0.155 - 0.987i$
Analytic cond. $0.478329$
Root an. cond. $0.478329$
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  + (−0.696 − 0.717i)2-s + (−0.602 − 0.798i)3-s + (−0.0307 + 0.999i)4-s + (0.332 + 0.943i)5-s + (−0.153 + 0.988i)6-s + (0.213 − 0.976i)7-s + (0.739 − 0.673i)8-s + (−0.273 + 0.961i)9-s + (0.445 − 0.895i)10-s + (0.969 − 0.243i)11-s + (0.816 − 0.577i)12-s + (0.739 + 0.673i)13-s + (−0.850 + 0.526i)14-s + (0.552 − 0.833i)15-s + (−0.998 − 0.0615i)16-s + (−0.153 − 0.988i)17-s + ⋯
L(s)  = 1  + (−0.696 − 0.717i)2-s + (−0.602 − 0.798i)3-s + (−0.0307 + 0.999i)4-s + (0.332 + 0.943i)5-s + (−0.153 + 0.988i)6-s + (0.213 − 0.976i)7-s + (0.739 − 0.673i)8-s + (−0.273 + 0.961i)9-s + (0.445 − 0.895i)10-s + (0.969 − 0.243i)11-s + (0.816 − 0.577i)12-s + (0.739 + 0.673i)13-s + (−0.850 + 0.526i)14-s + (0.552 − 0.833i)15-s + (−0.998 − 0.0615i)16-s + (−0.153 − 0.988i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.155 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.155 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(103\)
Sign: $0.155 - 0.987i$
Analytic conductor: \(0.478329\)
Root analytic conductor: \(0.478329\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{103} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 103,\ (0:\ ),\ 0.155 - 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5072015110 - 0.4333904807i\)
\(L(\frac12)\) \(\approx\) \(0.5072015110 - 0.4333904807i\)
\(L(1)\) \(\approx\) \(0.6311137131 - 0.3301776960i\)
\(L(1)\) \(\approx\) \(0.6311137131 - 0.3301776960i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad103 \( 1 \)
good2 \( 1 + (-0.696 - 0.717i)T \)
3 \( 1 + (-0.602 - 0.798i)T \)
5 \( 1 + (0.332 + 0.943i)T \)
7 \( 1 + (0.213 - 0.976i)T \)
11 \( 1 + (0.969 - 0.243i)T \)
13 \( 1 + (0.739 + 0.673i)T \)
17 \( 1 + (-0.153 - 0.988i)T \)
19 \( 1 + (-0.389 - 0.920i)T \)
23 \( 1 + (-0.273 - 0.961i)T \)
29 \( 1 + (0.650 - 0.759i)T \)
31 \( 1 + (0.445 + 0.895i)T \)
37 \( 1 + (0.0922 + 0.995i)T \)
41 \( 1 + (0.332 - 0.943i)T \)
43 \( 1 + (0.816 + 0.577i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (0.992 + 0.122i)T \)
59 \( 1 + (0.213 + 0.976i)T \)
61 \( 1 + (0.932 + 0.361i)T \)
67 \( 1 + (-0.952 - 0.303i)T \)
71 \( 1 + (0.650 + 0.759i)T \)
73 \( 1 + (-0.982 + 0.183i)T \)
79 \( 1 + (-0.982 - 0.183i)T \)
83 \( 1 + (-0.952 + 0.303i)T \)
89 \( 1 + (-0.850 + 0.526i)T \)
97 \( 1 + (-0.153 + 0.988i)T \)
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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

−29.60881183790652295449373163907, −28.42070422400346281892204652988, −27.91922126039197817671412505109, −27.31015131724595563463086356724, −25.76332459942718789685470175134, −25.06965690046935665439217640850, −24.00139832784753412699172786328, −22.91073479307735235863874104568, −21.712120889594854560176453702839, −20.661257153474976422268131482514, −19.48117284668329799253091762549, −17.95306650533432466392226185829, −17.30784724476381341858840158832, −16.29108317926612144365067104330, −15.44443259180231527833880973780, −14.44864872673061201516873716643, −12.640545643049927876439571117841, −11.38818304350074764212743439844, −10.05267675759020940309942485527, −9.088868368199085177250473360093, −8.27076874624456030114432637602, −6.14048754725439366281255749153, −5.59402069699954898549890622222, −4.21296674006912418363971648108, −1.46887222684878637771508002912, 1.11132400320711961466109563050, 2.580488003641404038378323277933, 4.25432643549321085250193546799, 6.55227881995540666946796758174, 7.124022928988896157885498957187, 8.6211843698317781117542629106, 10.17335792784497334549633772865, 11.133101164197886806794443537401, 11.80576602693029451981770447380, 13.4267537291101480571329253943, 14.06246193413050718352058105595, 16.261303271965227453573496563391, 17.28014657029970771517327420745, 18.00320995591027119019811185119, 18.98130706265351379427767379549, 19.81717844457458489825271907031, 21.19334466065449518485804690759, 22.34088408590319232095426027044, 23.08879822310167342508816585209, 24.50583792200904405857896605031, 25.67533758691681680624439337451, 26.6234288790596056564945689553, 27.58889013308488703993354191287, 28.78806570022772900258124570089, 29.58200158680727114078691803135

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