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Properties

Label 1-103-103.41-r0-0-0
Degree $1$
Conductor $103$
Sign $0.822 + 0.569i$
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  + (−0.908 − 0.417i)2-s + (0.739 + 0.673i)3-s + (0.650 + 0.759i)4-s + (−0.998 + 0.0615i)5-s + (−0.389 − 0.920i)6-s + (0.969 − 0.243i)7-s + (−0.273 − 0.961i)8-s + (0.0922 + 0.995i)9-s + (0.932 + 0.361i)10-s + (0.816 − 0.577i)11-s + (−0.0307 + 0.999i)12-s + (−0.273 + 0.961i)13-s + (−0.982 − 0.183i)14-s + (−0.779 − 0.626i)15-s + (−0.153 + 0.988i)16-s + (−0.389 + 0.920i)17-s + ⋯
L(s)  = 1  + (−0.908 − 0.417i)2-s + (0.739 + 0.673i)3-s + (0.650 + 0.759i)4-s + (−0.998 + 0.0615i)5-s + (−0.389 − 0.920i)6-s + (0.969 − 0.243i)7-s + (−0.273 − 0.961i)8-s + (0.0922 + 0.995i)9-s + (0.932 + 0.361i)10-s + (0.816 − 0.577i)11-s + (−0.0307 + 0.999i)12-s + (−0.273 + 0.961i)13-s + (−0.982 − 0.183i)14-s + (−0.779 − 0.626i)15-s + (−0.153 + 0.988i)16-s + (−0.389 + 0.920i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.822 + 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.822 + 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(103\)
Sign: $0.822 + 0.569i$
Analytic conductor: \(0.478329\)
Root analytic conductor: \(0.478329\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{103} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 103,\ (0:\ ),\ 0.822 + 0.569i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7815911821 + 0.2440931392i\)
\(L(\frac12)\) \(\approx\) \(0.7815911821 + 0.2440931392i\)
\(L(1)\) \(\approx\) \(0.8408152132 + 0.1192511482i\)
\(L(1)\) \(\approx\) \(0.8408152132 + 0.1192511482i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad103 \( 1 \)
good2 \( 1 + (-0.908 - 0.417i)T \)
3 \( 1 + (0.739 + 0.673i)T \)
5 \( 1 + (-0.998 + 0.0615i)T \)
7 \( 1 + (0.969 - 0.243i)T \)
11 \( 1 + (0.816 - 0.577i)T \)
13 \( 1 + (-0.273 + 0.961i)T \)
17 \( 1 + (-0.389 + 0.920i)T \)
19 \( 1 + (0.213 + 0.976i)T \)
23 \( 1 + (0.0922 - 0.995i)T \)
29 \( 1 + (0.552 + 0.833i)T \)
31 \( 1 + (0.932 - 0.361i)T \)
37 \( 1 + (-0.850 - 0.526i)T \)
41 \( 1 + (-0.998 - 0.0615i)T \)
43 \( 1 + (-0.0307 - 0.999i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (-0.952 - 0.303i)T \)
59 \( 1 + (0.969 + 0.243i)T \)
61 \( 1 + (-0.602 - 0.798i)T \)
67 \( 1 + (-0.696 + 0.717i)T \)
71 \( 1 + (0.552 - 0.833i)T \)
73 \( 1 + (0.445 + 0.895i)T \)
79 \( 1 + (0.445 - 0.895i)T \)
83 \( 1 + (-0.696 - 0.717i)T \)
89 \( 1 + (-0.982 - 0.183i)T \)
97 \( 1 + (-0.389 - 0.920i)T \)
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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

−29.876686432187697768563808449500, −28.38163976677337859918944747646, −27.38161638483488398673747366336, −26.81127080235130840244396715595, −25.44813092443886675586373934165, −24.669048132233424942584456801611, −23.97269327548947875791927311625, −22.815735025452654427944248629192, −20.77979760903962241975983258597, −19.924710455553876662096563020776, −19.28972475714131529867955145360, −18.00980373793584046839961173979, −17.4094330649015888850935724140, −15.55565138043539661203246700413, −15.08877264105796151546962444974, −13.88756565994744432119988408729, −12.0988047169347121672308887340, −11.333545227306773537815530419431, −9.57406828618200101748290919439, −8.46317125266511429813620268180, −7.68057311164400714301010364854, −6.77252770582006492660730940903, −4.83545350742589358484781185716, −2.81486259102884928535986806367, −1.202940320529788322772915683208, 1.77493264525356585381585466402, 3.49888257954276652427094747426, 4.39996324287882797976379611382, 6.9395350477778220275800533596, 8.273967836317709313034663929836, 8.71029586553536176448776056093, 10.268719719430096503746303016378, 11.203721401502253288218139091923, 12.176539428713502062912777312553, 14.096312974113144925806059260077, 15.0549722265279802711293072743, 16.278621243414442812783900495350, 17.05012652303282790971202175647, 18.70373612384745062645158260087, 19.443765824410767842167066754563, 20.33995037691684296028929378991, 21.23676684437785151360988233666, 22.240400912109100723974577770368, 24.0412574117577938308339316513, 24.88209164130662836353568376023, 26.329845238026507275802141062511, 26.91534152532274487002406939961, 27.5332075975496488474708084527, 28.53247138571512006992254821399, 30.10153168166861974939808034952

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