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Properties

Label 1-103-103.29-r0-0-0
Degree $1$
Conductor $103$
Sign $0.970 + 0.242i$
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  + (0.816 + 0.577i)2-s + (0.739 − 0.673i)3-s + (0.332 + 0.943i)4-s + (0.552 − 0.833i)5-s + (0.992 − 0.122i)6-s + (−0.696 + 0.717i)7-s + (−0.273 + 0.961i)8-s + (0.0922 − 0.995i)9-s + (0.932 − 0.361i)10-s + (−0.908 + 0.417i)11-s + (0.881 + 0.473i)12-s + (−0.273 − 0.961i)13-s + (−0.982 + 0.183i)14-s + (−0.153 − 0.988i)15-s + (−0.779 + 0.626i)16-s + (0.992 + 0.122i)17-s + ⋯
L(s)  = 1  + (0.816 + 0.577i)2-s + (0.739 − 0.673i)3-s + (0.332 + 0.943i)4-s + (0.552 − 0.833i)5-s + (0.992 − 0.122i)6-s + (−0.696 + 0.717i)7-s + (−0.273 + 0.961i)8-s + (0.0922 − 0.995i)9-s + (0.932 − 0.361i)10-s + (−0.908 + 0.417i)11-s + (0.881 + 0.473i)12-s + (−0.273 − 0.961i)13-s + (−0.982 + 0.183i)14-s + (−0.153 − 0.988i)15-s + (−0.779 + 0.626i)16-s + (0.992 + 0.122i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(103\)
Sign: $0.970 + 0.242i$
Analytic conductor: \(0.478329\)
Root analytic conductor: \(0.478329\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{103} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 103,\ (0:\ ),\ 0.970 + 0.242i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.835928188 + 0.2259014289i\)
\(L(\frac12)\) \(\approx\) \(1.835928188 + 0.2259014289i\)
\(L(1)\) \(\approx\) \(1.776207247 + 0.1956769197i\)
\(L(1)\) \(\approx\) \(1.776207247 + 0.1956769197i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad103 \( 1 \)
good2 \( 1 + (0.816 + 0.577i)T \)
3 \( 1 + (0.739 - 0.673i)T \)
5 \( 1 + (0.552 - 0.833i)T \)
7 \( 1 + (-0.696 + 0.717i)T \)
11 \( 1 + (-0.908 + 0.417i)T \)
13 \( 1 + (-0.273 - 0.961i)T \)
17 \( 1 + (0.992 + 0.122i)T \)
19 \( 1 + (-0.952 + 0.303i)T \)
23 \( 1 + (0.0922 + 0.995i)T \)
29 \( 1 + (-0.998 - 0.0615i)T \)
31 \( 1 + (0.932 + 0.361i)T \)
37 \( 1 + (-0.850 + 0.526i)T \)
41 \( 1 + (0.552 + 0.833i)T \)
43 \( 1 + (0.881 - 0.473i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (0.213 - 0.976i)T \)
59 \( 1 + (-0.696 - 0.717i)T \)
61 \( 1 + (-0.602 + 0.798i)T \)
67 \( 1 + (0.969 - 0.243i)T \)
71 \( 1 + (-0.998 + 0.0615i)T \)
73 \( 1 + (0.445 - 0.895i)T \)
79 \( 1 + (0.445 + 0.895i)T \)
83 \( 1 + (0.969 + 0.243i)T \)
89 \( 1 + (-0.982 + 0.183i)T \)
97 \( 1 + (0.992 - 0.122i)T \)
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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

−29.83421660209484490294342095400, −29.087469785901289085662850455648, −27.80964288397145968815676594690, −26.3606462757826349516801248653, −25.96957999042139715881667834982, −24.53028156749723533333138649771, −23.216801544015899296099846402093, −22.3334334471128777709958313284, −21.29651821948177011638848480738, −20.74504400290825564588510074342, −19.28068777629958684779855571122, −18.8258460766116207762206545491, −16.74786864451686370723372538333, −15.6184885029791784039596022818, −14.435531065937194306934784186126, −13.84271430829551925798559491236, −12.80029069558310165896608274451, −10.93297853507142478187529860614, −10.30574934285544644586744981512, −9.32456545716386255453582247502, −7.32327218988216373277633281294, −5.98666773943224196023195005803, −4.44565400803751308840887502412, −3.25605277588062147975426093165, −2.29879628888716817419129966158, 2.16225436584089091963668527483, 3.35597943475869338518028407079, 5.20393368474379409471365820612, 6.146765204618894660763940380415, 7.63926319806518157090364556223, 8.546660215566635710092339536270, 9.858249728646218694969440316533, 12.21545696004894817635787599695, 12.79307661208282806371938138554, 13.52093724617600982746621098019, 14.89110984053607060636320860124, 15.70689993557117791598154053925, 17.05648591670541769878758807456, 18.11589128045257635622058714532, 19.49656356082917134631171920164, 20.73005311848209648486356599143, 21.34895905537563466222678451034, 22.82217305069468161694801930031, 23.78247380638703776549337564211, 24.84075011486223030934067578904, 25.44515703634365575344924730467, 26.07529712816648149847026860593, 27.879660700747424014668295944932, 29.25953460426787982597140519535, 29.87416843708960156168349861174

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