↑

Properties

Label 1-103-103.58-r0-0-0
Degree $1$
Conductor $103$
Sign $0.806 + 0.591i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.881 + 0.473i)2-s + (−0.273 − 0.961i)3-s + (0.552 + 0.833i)4-s + (−0.153 + 0.988i)5-s + (0.213 − 0.976i)6-s + (0.816 + 0.577i)7-s + (0.0922 + 0.995i)8-s + (−0.850 + 0.526i)9-s + (−0.602 + 0.798i)10-s + (−0.0307 − 0.999i)11-s + (0.650 − 0.759i)12-s + (0.0922 − 0.995i)13-s + (0.445 + 0.895i)14-s + (0.992 − 0.122i)15-s + (−0.389 + 0.920i)16-s + (0.213 + 0.976i)17-s + ⋯
L(s)  = 1  + (0.881 + 0.473i)2-s + (−0.273 − 0.961i)3-s + (0.552 + 0.833i)4-s + (−0.153 + 0.988i)5-s + (0.213 − 0.976i)6-s + (0.816 + 0.577i)7-s + (0.0922 + 0.995i)8-s + (−0.850 + 0.526i)9-s + (−0.602 + 0.798i)10-s + (−0.0307 − 0.999i)11-s + (0.650 − 0.759i)12-s + (0.0922 − 0.995i)13-s + (0.445 + 0.895i)14-s + (0.992 − 0.122i)15-s + (−0.389 + 0.920i)16-s + (0.213 + 0.976i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.806 + 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.806 + 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(103\)
Sign: $0.806 + 0.591i$
Analytic conductor: \(0.478329\)
Root analytic conductor: \(0.478329\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{103} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 103,\ (0:\ ),\ 0.806 + 0.591i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.459510942 + 0.4780147119i\)
\(L(\frac12)\) \(\approx\) \(1.459510942 + 0.4780147119i\)
\(L(1)\) \(\approx\) \(1.478378701 + 0.3168895252i\)
\(L(1)\) \(\approx\) \(1.478378701 + 0.3168895252i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad103 \( 1 \)
good2 \( 1 + (0.881 + 0.473i)T \)
3 \( 1 + (-0.273 - 0.961i)T \)
5 \( 1 + (-0.153 + 0.988i)T \)
7 \( 1 + (0.816 + 0.577i)T \)
11 \( 1 + (-0.0307 - 0.999i)T \)
13 \( 1 + (0.0922 - 0.995i)T \)
17 \( 1 + (0.213 + 0.976i)T \)
19 \( 1 + (0.969 - 0.243i)T \)
23 \( 1 + (-0.850 - 0.526i)T \)
29 \( 1 + (-0.779 - 0.626i)T \)
31 \( 1 + (-0.602 - 0.798i)T \)
37 \( 1 + (-0.982 - 0.183i)T \)
41 \( 1 + (-0.153 - 0.988i)T \)
43 \( 1 + (0.650 + 0.759i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (-0.696 - 0.717i)T \)
59 \( 1 + (0.816 - 0.577i)T \)
61 \( 1 + (0.739 - 0.673i)T \)
67 \( 1 + (-0.908 - 0.417i)T \)
71 \( 1 + (-0.779 + 0.626i)T \)
73 \( 1 + (0.932 + 0.361i)T \)
79 \( 1 + (0.932 - 0.361i)T \)
83 \( 1 + (-0.908 + 0.417i)T \)
89 \( 1 + (0.445 + 0.895i)T \)
97 \( 1 + (0.213 - 0.976i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−29.63549003065921328855222227704, −28.57888610759384022829772240118, −27.91366065007938847666110860435, −26.96361537505588867483158667409, −25.43021844127609572752358587681, −24.11845988754092346834207365964, −23.422568860314473809767816033381, −22.36114386155535518115943008069, −21.18471073233368607415588669653, −20.53550816819170686878975751700, −19.91586393435148219374729197863, −17.990032037899787010201778171497, −16.61522991003566601850461765828, −15.84926816759543391421984094751, −14.59463682583671149833868674443, −13.70172986815301601938298516656, −12.101963634988602382979706803473, −11.50221215585699905140167903808, −10.14048761514976736197195759561, −9.162397832462977359577512935050, −7.2838714742561263232885368053, −5.383513782050031818193893617802, −4.68628450271261491002733522433, −3.725517216334910114888077293027, −1.60384453237630171147665237900, 2.19690756501804821373587341865, 3.45353175157909318029477738497, 5.49213388650768617126103963080, 6.17225248642457052431565979402, 7.60172981297797625514411227386, 8.25576100248510348273446923781, 10.88715591178762123066036379019, 11.60490077742323852809573783073, 12.75021877180308680873261937055, 13.93331870615065642051886483498, 14.69711778777261183058444212792, 15.85662558857801937710258906838, 17.348993235972043706326108709491, 18.16973725431805997283937038612, 19.25829906302513942320083679403, 20.70823667063934252805101376600, 22.090799303062456203016957341588, 22.56766950989794257985542834847, 23.94623460664268002722081260387, 24.37850974930489481570909458775, 25.530306335395619403740131810771, 26.50272512086190714929055644518, 27.925804962771984032467939296354, 29.38562207887122667920265047837, 30.19518789103197815632409945363

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