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Properties

Label 1-1872-1872.587-r1-0-0
Degree $1$
Conductor $1872$
Sign $0.0693 + 0.997i$
Analytic cond. $201.174$
Root an. cond. $201.174$
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  + (0.5 − 0.866i)5-s + (−0.866 − 0.5i)7-s − 11-s + (−0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + (−0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s − i·29-s + (−0.866 − 0.5i)31-s + (−0.866 + 0.5i)35-s + (0.5 − 0.866i)37-s + (−0.866 + 0.5i)41-s + (−0.866 − 0.5i)43-s + (0.866 − 0.5i)47-s + (0.5 + 0.866i)49-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)5-s + (−0.866 − 0.5i)7-s − 11-s + (−0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + (−0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s − i·29-s + (−0.866 − 0.5i)31-s + (−0.866 + 0.5i)35-s + (0.5 − 0.866i)37-s + (−0.866 + 0.5i)41-s + (−0.866 − 0.5i)43-s + (0.866 − 0.5i)47-s + (0.5 + 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0693 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0693 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1872\)    =    \(2^{4} \cdot 3^{2} \cdot 13\)
Sign: $0.0693 + 0.997i$
Analytic conductor: \(201.174\)
Root analytic conductor: \(201.174\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1872} (587, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1872,\ (1:\ ),\ 0.0693 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2188228083 - 0.2041353324i\)
\(L(\frac12)\) \(\approx\) \(-0.2188228083 - 0.2041353324i\)
\(L(1)\) \(\approx\) \(0.7194009822 - 0.3268882049i\)
\(L(1)\) \(\approx\) \(0.7194009822 - 0.3268882049i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 - T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 - iT \)
31 \( 1 + (-0.866 - 0.5i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (-0.866 + 0.5i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 - iT \)
59 \( 1 - T \)
61 \( 1 + (0.866 + 0.5i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.866 - 0.5i)T \)
73 \( 1 - iT \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 + (0.866 + 0.5i)T \)
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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

−20.47244848321565678594852117818, −19.63894301851929528473423466092, −18.734394717658325518890900600489, −18.57645641004610678235657306247, −17.57270093764538669099123367352, −16.924170055536181498014072164137, −15.8957304416574433723236662757, −15.35382358801902716755391372629, −14.70824439111859757020027569756, −13.70169068543278013286106797807, −13.161337982069505608478570932775, −12.42694616086944128659513979668, −11.500082482319196807884420530189, −10.56457980167202625437888875312, −10.105455102348908326363094380164, −9.362670452073297473876161231345, −8.360795763699015647584556175, −7.580835128652042998835558120680, −6.603061378326191575688897722634, −6.00089802944225351102574298481, −5.37698315243237222191068361111, −4.04268784043736750983789038843, −3.230643822041061362608886402827, −2.43411310434234763304272205649, −1.66871980114561993803266352730, 0.07654122756957994088967954110, 0.59745336746650504150975463404, 1.985283939106409970039916696748, 2.73354570517933385457745951166, 3.82931689981633942443155334975, 4.77377820633749612621686024056, 5.37315395599035468082091202201, 6.39218289785057164880440562940, 7.07397522495440550906050948675, 8.04766034075434001638997254113, 8.92674700801896381875952503703, 9.50294566915441238608665698516, 10.34775873048980500039912197187, 10.97119035711788819015482438134, 12.12987364761392979283222096017, 12.8544447786099986877204406175, 13.33302559406009180271696176125, 13.93088669669155103337532187649, 15.055279440703380342606792444957, 15.891116291042261892529893558418, 16.45297417168559610295090392162, 16.97922879608972563710164931616, 18.052302282809877517836144734443, 18.40784228192121547264095260624, 19.59886444756418205676638755966

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