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Properties

Label 1-103-103.38-r0-0-0
Degree $1$
Conductor $103$
Sign $0.102 + 0.994i$
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  + (−0.998 + 0.0615i)2-s + (−0.850 + 0.526i)3-s + (0.992 − 0.122i)4-s + (0.213 + 0.976i)5-s + (0.816 − 0.577i)6-s + (0.650 − 0.759i)7-s + (−0.982 + 0.183i)8-s + (0.445 − 0.895i)9-s + (−0.273 − 0.961i)10-s + (0.552 + 0.833i)11-s + (−0.779 + 0.626i)12-s + (−0.982 − 0.183i)13-s + (−0.602 + 0.798i)14-s + (−0.696 − 0.717i)15-s + (0.969 − 0.243i)16-s + (0.816 + 0.577i)17-s + ⋯
L(s)  = 1  + (−0.998 + 0.0615i)2-s + (−0.850 + 0.526i)3-s + (0.992 − 0.122i)4-s + (0.213 + 0.976i)5-s + (0.816 − 0.577i)6-s + (0.650 − 0.759i)7-s + (−0.982 + 0.183i)8-s + (0.445 − 0.895i)9-s + (−0.273 − 0.961i)10-s + (0.552 + 0.833i)11-s + (−0.779 + 0.626i)12-s + (−0.982 − 0.183i)13-s + (−0.602 + 0.798i)14-s + (−0.696 − 0.717i)15-s + (0.969 − 0.243i)16-s + (0.816 + 0.577i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.102 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.102 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(103\)
Sign: $0.102 + 0.994i$
Analytic conductor: \(0.478329\)
Root analytic conductor: \(0.478329\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{103} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 103,\ (0:\ ),\ 0.102 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4007477227 + 0.3615147659i\)
\(L(\frac12)\) \(\approx\) \(0.4007477227 + 0.3615147659i\)
\(L(1)\) \(\approx\) \(0.5540443139 + 0.2272902718i\)
\(L(1)\) \(\approx\) \(0.5540443139 + 0.2272902718i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad103 \( 1 \)
good2 \( 1 + (-0.998 + 0.0615i)T \)
3 \( 1 + (-0.850 + 0.526i)T \)
5 \( 1 + (0.213 + 0.976i)T \)
7 \( 1 + (0.650 - 0.759i)T \)
11 \( 1 + (0.552 + 0.833i)T \)
13 \( 1 + (-0.982 - 0.183i)T \)
17 \( 1 + (0.816 + 0.577i)T \)
19 \( 1 + (-0.0307 + 0.999i)T \)
23 \( 1 + (0.445 + 0.895i)T \)
29 \( 1 + (-0.952 - 0.303i)T \)
31 \( 1 + (-0.273 + 0.961i)T \)
37 \( 1 + (0.932 + 0.361i)T \)
41 \( 1 + (0.213 - 0.976i)T \)
43 \( 1 + (-0.779 - 0.626i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (0.881 - 0.473i)T \)
59 \( 1 + (0.650 + 0.759i)T \)
61 \( 1 + (0.0922 - 0.995i)T \)
67 \( 1 + (0.332 - 0.943i)T \)
71 \( 1 + (-0.952 + 0.303i)T \)
73 \( 1 + (0.739 + 0.673i)T \)
79 \( 1 + (0.739 - 0.673i)T \)
83 \( 1 + (0.332 + 0.943i)T \)
89 \( 1 + (-0.602 + 0.798i)T \)
97 \( 1 + (0.816 - 0.577i)T \)
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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

−29.38161806784294215948704841029, −28.31998128785877801498971099323, −27.83104268717481380961295610371, −26.813030635798207825384848387980, −25.07844995537719301869342475087, −24.572868699181557278752951201863, −23.86801336739796064425047299636, −22.021857005502232148681790642533, −21.209376728002944688683218663868, −19.88503523971533353027652761607, −18.81873601219692936087804295667, −17.93288971262761007057665022005, −16.84420000081152550625762488121, −16.40893586977328037243668610793, −14.802077603016608093665909267, −12.995474654161402747888529497089, −11.8876159280119197875143086914, −11.29236646238484634232801682234, −9.66000110692387289869369288269, −8.61404639585315002375234342363, −7.46231436767832497274564958976, −6.03673646310682584385104587241, −4.97135226335434141684555718562, −2.29267145713070759391413833348, −0.89393540873360737706986474997, 1.644689337001229657034060670917, 3.67510469112723308976373216263, 5.47763695272870509016984788656, 6.83736343095629797177323480548, 7.66108330044774239210247719342, 9.66121951936253203716658039184, 10.269338665802602214946736327041, 11.234824277561646937347648372335, 12.25296187497568394125354991419, 14.54624246919422253706171891077, 15.12456173848593166086205001564, 16.73102436708378337914196934359, 17.33420377085719828643187745553, 18.14710703239398459143224938177, 19.388383233810179797255786623738, 20.637590785805296008371030657291, 21.61593572564597993161849373312, 22.83182593799112569999102487753, 23.778797579009930887156792480872, 25.159887189608261849629447773528, 26.22468965556667220284669538401, 27.253762514479049951921998136997, 27.5606581569099181341747965847, 29.01624503197790273519598649670, 29.750451187874305014460627916120

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