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Properties

Label 1-2183-2183.39-r0-0-0
Degree $1$
Conductor $2183$
Sign $0.658 - 0.752i$
Analytic cond. $10.1378$
Root an. cond. $10.1378$
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  + (−0.505 − 0.863i)2-s + (−0.734 − 0.678i)3-s + (−0.489 + 0.871i)4-s + (−0.977 + 0.209i)5-s + (−0.214 + 0.976i)6-s + (−0.692 + 0.721i)7-s + (0.999 − 0.0180i)8-s + (0.0781 + 0.996i)9-s + (0.674 + 0.738i)10-s + (0.197 − 0.980i)11-s + (0.951 − 0.307i)12-s + (0.996 + 0.0781i)13-s + (0.972 + 0.232i)14-s + (0.859 + 0.510i)15-s + (−0.520 − 0.853i)16-s + (−0.238 − 0.971i)17-s + ⋯
L(s)  = 1  + (−0.505 − 0.863i)2-s + (−0.734 − 0.678i)3-s + (−0.489 + 0.871i)4-s + (−0.977 + 0.209i)5-s + (−0.214 + 0.976i)6-s + (−0.692 + 0.721i)7-s + (0.999 − 0.0180i)8-s + (0.0781 + 0.996i)9-s + (0.674 + 0.738i)10-s + (0.197 − 0.980i)11-s + (0.951 − 0.307i)12-s + (0.996 + 0.0781i)13-s + (0.972 + 0.232i)14-s + (0.859 + 0.510i)15-s + (−0.520 − 0.853i)16-s + (−0.238 − 0.971i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2183\)    =    \(37 \cdot 59\)
Sign: $0.658 - 0.752i$
Analytic conductor: \(10.1378\)
Root analytic conductor: \(10.1378\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2183} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2183,\ (0:\ ),\ 0.658 - 0.752i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4854690367 - 0.2203084290i\)
\(L(\frac12)\) \(\approx\) \(0.4854690367 - 0.2203084290i\)
\(L(1)\) \(\approx\) \(0.4599127430 - 0.2141428512i\)
\(L(1)\) \(\approx\) \(0.4599127430 - 0.2141428512i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
59 \( 1 \)
good2 \( 1 + (-0.505 - 0.863i)T \)
3 \( 1 + (-0.734 - 0.678i)T \)
5 \( 1 + (-0.977 + 0.209i)T \)
7 \( 1 + (-0.692 + 0.721i)T \)
11 \( 1 + (0.197 - 0.980i)T \)
13 \( 1 + (0.996 + 0.0781i)T \)
17 \( 1 + (-0.238 - 0.971i)T \)
19 \( 1 + (0.226 + 0.973i)T \)
23 \( 1 + (0.995 - 0.0901i)T \)
29 \( 1 + (-0.941 + 0.336i)T \)
31 \( 1 + (-0.998 - 0.0541i)T \)
41 \( 1 + (0.996 - 0.0841i)T \)
43 \( 1 + (0.319 - 0.947i)T \)
47 \( 1 + (-0.530 + 0.847i)T \)
53 \( 1 + (0.609 + 0.792i)T \)
61 \( 1 + (-0.937 + 0.347i)T \)
67 \( 1 + (-0.324 + 0.945i)T \)
71 \( 1 + (-0.788 + 0.614i)T \)
73 \( 1 + (-0.725 - 0.687i)T \)
79 \( 1 + (-0.0481 - 0.998i)T \)
83 \( 1 + (-0.0300 + 0.999i)T \)
89 \( 1 + (-0.167 - 0.985i)T \)
97 \( 1 + (-0.284 + 0.958i)T \)
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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

−19.82053935619432771733370096353, −19.12012312353976239037948270882, −18.136625473728710937418352707431, −17.55936280996507201979192199910, −16.67140344811258974299511985623, −16.464768568331039725494434411571, −15.42313803891766215243890656695, −15.31051642512727869069583447113, −14.4314909351743904294381127498, −13.07296768316476634674920511160, −12.83046371111022039178520780434, −11.4582699361248517549489185598, −10.937818328740065125215311812501, −10.28476769642555115611473074129, −9.31982776904525565951637513049, −8.930955348871132183612652785232, −7.77978068430728138017858233063, −7.06519432413072747803846138831, −6.497455951528842247639135415732, −5.60035015872898811608322485752, −4.64145734064935833221454387713, −4.11316081090247316938620451161, −3.37105908607405808696961480812, −1.44761633090505235398439708096, −0.483248151128687699423671259, 0.58890161466644635186144771009, 1.4906905061927621303747009936, 2.70722580772468431478135891741, 3.348008417528715749069810834895, 4.1869546711940975378315399262, 5.37565676185656919420107719392, 6.12934830911422324496811932979, 7.15229952727058092636120205330, 7.6931834732927389354591259271, 8.74453049687219282825759165631, 9.08240421597938054453334069653, 10.391917263782899807385784647791, 11.10139204527229666468484352301, 11.4924375580304092724112899589, 12.18720284059173085639888985343, 12.85358399206588291143442368885, 13.473450508536058860782879444499, 14.40193389807846591298752981100, 15.67900826197790724751342126799, 16.41988116403467878801606829628, 16.56022378432077875103864914802, 17.853532786795932705875914315342, 18.55464807332680317027193612141, 18.8227001543594520649048521351, 19.33728581777173934377189097545

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