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Properties

Label 1-103-103.13-r0-0-0
Degree $1$
Conductor $103$
Sign $0.982 - 0.185i$
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  + (0.932 + 0.361i)2-s + (−0.982 − 0.183i)3-s + (0.739 + 0.673i)4-s + (−0.273 − 0.961i)5-s + (−0.850 − 0.526i)6-s + (0.445 − 0.895i)7-s + (0.445 + 0.895i)8-s + (0.932 + 0.361i)9-s + (0.0922 − 0.995i)10-s + (0.932 + 0.361i)11-s + (−0.602 − 0.798i)12-s + (0.445 − 0.895i)13-s + (0.739 − 0.673i)14-s + (0.0922 + 0.995i)15-s + (0.0922 + 0.995i)16-s + (−0.850 + 0.526i)17-s + ⋯
L(s)  = 1  + (0.932 + 0.361i)2-s + (−0.982 − 0.183i)3-s + (0.739 + 0.673i)4-s + (−0.273 − 0.961i)5-s + (−0.850 − 0.526i)6-s + (0.445 − 0.895i)7-s + (0.445 + 0.895i)8-s + (0.932 + 0.361i)9-s + (0.0922 − 0.995i)10-s + (0.932 + 0.361i)11-s + (−0.602 − 0.798i)12-s + (0.445 − 0.895i)13-s + (0.739 − 0.673i)14-s + (0.0922 + 0.995i)15-s + (0.0922 + 0.995i)16-s + (−0.850 + 0.526i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(103\)
Sign: $0.982 - 0.185i$
Analytic conductor: \(0.478329\)
Root analytic conductor: \(0.478329\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{103} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 103,\ (0:\ ),\ 0.982 - 0.185i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.340553687 - 0.1255394197i\)
\(L(\frac12)\) \(\approx\) \(1.340553687 - 0.1255394197i\)
\(L(1)\) \(\approx\) \(1.345039795 + 0.02307862588i\)
\(L(1)\) \(\approx\) \(1.345039795 + 0.02307862588i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad103 \( 1 \)
good2 \( 1 + (0.932 + 0.361i)T \)
3 \( 1 + (-0.982 - 0.183i)T \)
5 \( 1 + (-0.273 - 0.961i)T \)
7 \( 1 + (0.445 - 0.895i)T \)
11 \( 1 + (0.932 + 0.361i)T \)
13 \( 1 + (0.445 - 0.895i)T \)
17 \( 1 + (-0.850 + 0.526i)T \)
19 \( 1 + (-0.982 + 0.183i)T \)
23 \( 1 + (0.932 - 0.361i)T \)
29 \( 1 + (-0.273 - 0.961i)T \)
31 \( 1 + (0.0922 + 0.995i)T \)
37 \( 1 + (-0.602 - 0.798i)T \)
41 \( 1 + (-0.273 + 0.961i)T \)
43 \( 1 + (-0.602 + 0.798i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.982 + 0.183i)T \)
59 \( 1 + (0.445 + 0.895i)T \)
61 \( 1 + (-0.850 + 0.526i)T \)
67 \( 1 + (0.445 + 0.895i)T \)
71 \( 1 + (-0.273 + 0.961i)T \)
73 \( 1 + (-0.273 + 0.961i)T \)
79 \( 1 + (-0.273 - 0.961i)T \)
83 \( 1 + (0.445 - 0.895i)T \)
89 \( 1 + (0.739 - 0.673i)T \)
97 \( 1 + (-0.850 - 0.526i)T \)
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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

−29.86289492611271307049813464142, −29.01672661438062969065116870504, −27.92778644590292232035205681590, −27.12325611444352850773965947768, −25.49488779743814815607701695331, −24.24415455639571481659589661141, −23.47347445688113011268291159976, −22.24103988203167578402672132229, −21.970495661376434446831120585781, −20.86456670040305171348191141943, −19.16776140585907740671356544244, −18.514249695159157062858393838, −17.03375553464722833310520520415, −15.65498213697808067937791428436, −14.95693380347946051114519934224, −13.73951530961080411731653743231, −12.22901302614469841349357298699, −11.37142790416831496872518120690, −10.86452735187230545468575527994, −9.19978802462980128093833179212, −6.898584034953767824782839504802, −6.225307160822097315057220889606, −4.86471915320739326014574861527, −3.65029729220500836175369679545, −1.94334165267772376428100897949, 1.447318542377203096522068272384, 4.043600652493082198825920506449, 4.744711926875166131909405223680, 6.07293033923323012389855280882, 7.19698033786730945965833238464, 8.457921797188813190593528417119, 10.579661406673558148886605877503, 11.5453430177327906620604262747, 12.67287944776634611482061181580, 13.32434359465484189959212911735, 14.8854455953846667443419345904, 16.01345334328743161130627262007, 17.10752495631728349241963195141, 17.458185834656570989159421098631, 19.59995835912982775147377016650, 20.59050243078119386175419301094, 21.59168136840928265975023272944, 22.92604485566403056942952110121, 23.356107324709102454711764692343, 24.45077523960628292545273339663, 25.0781713757837384600853174654, 26.81796314595269343700592433104, 27.84793559732764099054750223181, 28.86343099651964684536288352979, 30.06395349009914194843240913812

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